Create a piecewise linear function and sketch a graph for

If r is measured in inches, s in pounds, and t in minutes, identify the units for the following average rates of change: a. change in r change in s b. change in t change in r c. change in s change in r

Assume that R is measured in dollars, S in ounces, T in dollars per ounce, and V in ounces per dollar. Write a product of two of these terms whose resulting units will be: a. Dollars b. Ounces

Your cars gas tank is full and you take a trip. You travel 212 miles, then you fill your gas tank up again and it takes 10.8 gallons. If your change in distance is 212 miles and your change in gallons is 10.8, what is the average rate of change of gasoline used, measured in miles per gallon?

The gas gauge on your car is broken, but you know that the car averages 22 miles per gallon. You fill your 15.5-gallon gas tank and tell your friend, I can travel 300 miles before I need to fill up the tank again. Why is this true?

The consumption of margarine (in pounds per person) decreased from 10.9 in 1990 to 4.6 in 2006. What was the annual average rate of change? (Source: www.census.gov)

The percentage of people who own homes in the United States has gone from 65.5% in 1980 to 67.6% in 2009. What is the average rate of change in percentage points per year?

The accompanying table shows females SAT scores in 2000 and 2008. Find the average rate of change: a. In the math scores from 2000 to 2008 b. In the verbal scores from 2000 to 2008

a. In 1992 the aerospace industry showed a net loss (negative profit) of $1.84 billion. In 2002 the industry had a net profit of $8.97 billion. Find the average annual rate of change in net profits from 1992 to 2002. b. In 2007, aerospace industry net profits were $16.9 billion. Find the average rate of change in net profits: i. From 1992 to 2007 ii. From 2002 to 2007

The U.S. Bureau of the Census provided the following data concerning computer use in all U.S. elementary and secondary schools. a. Find the average rate of change from 1985 to 2005 in: i. The number of computers being used ii. The number of students per computer b. Find the average rate of change from 2005 to 2006 in: i. The number of computers being used ii. The number of students per computer c. Using the annual average rate of change between 2005 and 2006, in how many years after 2005 would there be a projection of two students per computer. d. Look online to see if you can find more up-to-date data.

According to the U.S. Bureau of the Census, the percentage of persons 25 years old and over completing 4 or more years of college was 4.6 in 1940 and 29.4 in 2008. a. Plot the data, labeling both axes and the coordinates of the points. b. Calculate the average rate of change in percentage points per year. c. Write a topic sentence summarizing what you think is the central idea to be drawn from these data.

According to a 2010 Kaiser Family Foundation report, during 2009 kids spent almost every waking hour outside of school using electronic devices such as iPhones, computers, or televisions. Those aged 8 to 18 spent an average seven and a half hours a day in 2009 versus six and a half hours in 2004. Calculate the annual average rate of change of electronic use for kids from 2004 to 2009.

The following graph was produced by the U.S. Department of Transportation to report actual and predicted highway crash fatalities (up until 2020). The left-hand axis shows the number of highway fatalities and the right-hand axis the percent of all crashes that were fatal. a. Estimate the number of fatalities (y1) in 1965 and 1995 and calculate the annual average rate of change. b. Estimate the number of fatalities (y1) in 1995 and those predicted to 2020. Calculate the annual average rate of change. c. The fatality rate (y2) between 1995 and 2020 is predicted to be flat ( just below 2%), while the number of fatalities is expected to increase. How could that be?

Use the information in the accompanying table to answer the following questions. a. What was the average rate of change (in percentage points per year) of completion of 4 years of high school from 1940 to 2008 for whites? For blacks? For Asian/Pacific Islanders? For all? b. If these rates continue, what percentages of whites, of blacks, of Asian/Pacific Islanders, and of all will have finished 4 years of high school in the year 2010? Check the Internet to see if your predictions are accurate. c. If these rates continue, in what year will 100% of whites have completed 4 years of high school or more? In what year 100% of blacks? In what year 100% of Asian/Pacific Islanders? Do these projections make sense? d. Write a 60-second summary describing the key elements in the high school completion data. Include rates of change and possible projections for the near future.

The data show U.S. consumption and exports of cigarettes. a. Calculate the average rates of change in U.S. cigarette consumption from 1960 to 1980, from 1980 to 2007, and from 1960 to 2007. b. Compute the average rate of change for cigarette exports from 1960 to 2007. Does this give an accurate image of cigarette exports? c. The total number of cigarettes consumed in the United States in 1960 was 484 billion, very close to the number consumed in 1995, 487 billion. Does that mean smoking was as popular in 1995 as it was in 1960? Explain your answer. d. Write a paragraph summarizing what the data tell you about the consumption and exports of cigarettes since 1960, including average rates of change.

Examine the accompanying table on life expectancy. a. What group had the highest life expectancy in 1900? In 2010? What group had the lowest life expectancy in 1900? In 2010? b. Which group had the largest average rate of change in life expectancy between 1900 and 2010? c. Write a short summary of the patterns in U.S. life expectancy from 1900 to 2010 using average rates of change to support your points.

The table below shows U.S. marital status a. Calculate the average rate of change in the number of unmarried males between 1950 and 2008. Interpret your results. b. Calculate the average rate of change in the number of unmarried females between 1950 and 2008. Interpret your results. c. Compare the two results. d. What does this tell you, if anything, about the percentages of unmarried males and females?

Calculate the average rate of change between adjacent points for the following function. (The first few are done for you.) a. Is the function (x) increasing, decreasing, or constant throughout? b. Is the average rate of change increasing, decreasing, or constant throughout?

Calculate the average rate of change between adjacent points for the following function. The first one is done for you. a. Is the function (x) increasing, decreasing, or constant throughout? b. Is the average rate of change increasing, decreasing, or constant throughout?

The accompanying table shows the number of registered motor vehicles in the United States. a. Fill in the third column in the table. b. During which decade was the average rate of change the smallest? c. During which decade was the average rate of change the largest? d. Write a paragraph describing the change in registered motor vehicles between 1970 and 2010.

(Graphing program optional.) The accompanying table indicates the number of juvenile arrests (in thousands) in the United States for aggravated assault. a. Fill in the third column in the table by calculating the annual average rate of change. b. Graph the annual average rate of change versus time. c. During what 5-year period was the annual average rate of change the largest? d. Describe the change in juvenile arrests during these years by referring both to the number and to the annual average rate of change.

Calculate the average rate of change between adjacent points for the following functions and place the values in a third column in each table. (The first entry is n.a.) a. Are the functions (x) and g(x) increasing, decreasing, or constant throughout? b. Is the average rate of change of each function increasing, decreasing, or constant throughout?

Calculate the average rate of change between adjacent points for each of the functions in Tables AD and place the values in a third column in each table. (The first entry is n.a.) Then for each function decide which statement best describes it. a. As x increases, the function increases at a constant rate. b. As x increases, the function increases at an increasing rate. c. As x increases, the function decreases at a constant rate. d. As x increases, the function decreases at a decreasing rate.

Each of the following functions has a graph that is increasing. If you calculated the average rate of change between sequential equal-size intervals, which function can be said to have an average rate of change that is: a. Constant? b. Increasing? c. Decreasing?

Match each data table with its graph.

Refer to the first two data tables (A and B) in Exercise 8. Insert a third column in each table and label the column average rate of change. (Again the first entry is n.a.) a. Calculate the average rate of change over adjacent data points. b. Identify whether the table represents an average rate of change that is constant, increasing, or decreasing. c. Explain how you could tell this by looking at the corresponding graph in Exercise 8.

Following are data on the U.S. population over the time period 18301930 (extracted from Table 2.1). a. Fill in the missing parts of the table. b. Which 20-year interval experienced the largest average rate of change in population? c. Which 20-year interval experienced the smallest average rate of change in population?

(Technology recommended.) The accompanying data give a picture of the two major methods of news communication in the United States. (See also Excel or graph link files NEWPRINT and ONAIRTV.) a. Use the U.S. population numbers from Table 2.1 (at the beginning of this chapter) to calculate and compare the number of copies of newspapers per person in 1920 and in 2000. b. Create a table that displays the annual average rate of change in TV stations for each decade between 1950 and 2000 and the 8-yr period between 2000 and 2008. Create a similar table that displays the annual average rate of change in newspapers published for the same period. Graph both of the results. c. If new TV stations continue to come into existence at the same rate as from 2000 to 2008, how many would there be in the year 2010? If possible, check your predictions on the Internet. Is your estimate too high or too low? d. What trends do you see in the dissemination of news as reflected in these data?

Find the slope of a straight line that goes through: a. (5, 6) and (2, 3) b. (5, 6) and (2, 3) 2.

Find the slope of each line using the points where the graph intersects the x and y axes.

Find the slope of each line using the points where the line crosses the x- or y-axis.

Examine the line segments A, B, and C. a. Which line segment has a slope that is positive? That is negative? That is zero? b. Calculate the exact slope for each line segment A, B, and C.

Given the following graph: a. Estimate the slope for each line segment AF . b. Which line segment is the steepest? c. Which line segment has a slope of zero?

Plot each pair of points and calculate the slope of the line that passes through them. a. (3, 5) and (8, 15) d. (2, 6) and (2, 6) b. (1, 4) and (7, 0) e. (4, 3) and (2, 3) c. (5, 9) and (5, 9) 1. Fi

The following problems represent calculations of the slopes of different lines. Solve for the variable in each equation. a. 150 2 75 20 2 10 5 m b. 70 2 y0 2 8 5 0.5 c. 182 2 150 28 2 x 5 4 d. 6 2 0 x 2 10 5 0.6

Find the slope m of the line through the points (0, b) and (x, y), then solve the equation for y.

Find the value of t if m is the slope of the line that passes through the given points. a. (3, t) and (2, 1), m 4 b. (5,6) and (t,9), m = 2/3

a. Find the value of x so that the slope of the line through (x, 5) and (4, 2) is . b. Find the value of y so that the slope of the line through (1, 3) and (4, y) is 2. c. Find the value of y so that the slope of the line through (2, 3) and (5, y) is 0. d. Find the value of x so that the slope of the line through (2, 2) and (x, 10) is 2. e. Find the value of y so that the slope of the line through (100, 10) and (0, y) is . f. Find at least one set of values for x and y so that the slope of the line through (5, 8) and (x, y) is 0. 11.

Points that lie on the same straight line are said to be collinear. Determine if the following points are collinear. a. (2, 3), (4, 7), and (8, 15) b. (3, 1), (2, 4), and (7, 8)

a. Find the slope of the line through each of the following pairs of points. i. (1, 4) and (2, 4) ii. (7, 3) and (7, 3) iii. (2, 6) and (5, 6) b. Summarize your findings. 13. Gr

Graph a line through each pair of points and then calculate its slope. a. The origin and (6, 2) b. The origin and (4, 7)

Find some possible values of the y-coordinates for the points (3, y1) and (6, y2) such that the slope m 0.

Calculate the slope of the line passing through each of the following pairs of points. a. and b. and c. (0, b) and (b, 0) d. What do these pairs of points and slopes all have in common?

Which pairs of points produce a line with a negative slope? a. (5, 5) and (3, 3) d. (4, 3) and (12, 0) b. (2, 6) and (1, 4) e. (0, 3) and (4, 10) c. (3, 7) and (3, 7) f. (4, 2) and (6, 2) (23 2

In the previous exercise, which pairs of points produce a line with a positive slope?

A study on numerous streams examined the effects of a warming climate. It found an increase in water temperature of about 0.7C for every 1C increase in air temperature. a. Find the rate of change in water temperature with respect to air temperature. What are the units? b. If the air temperature increased by 5C, by how much would you expect the stream temperature to increase?

Handicapped Vietnam veterans successfully lobbied for improvements in the architectural standards for wheelchair access to public buildings. a. The old standard was a 1-foot rise for every 10 horizontal feet. What would the slope be for a ramp built under this standard? b. The new standard is a 1-foot rise for every 12 horizontal feet. What would the slope of a ramp be under this standard? c. If the front door is 3 feet above the ground, how long would the handicapped ramp be using the old standard? Using the new?

The following graph shows the relationship between education and health in the United States. a. Estimate the percentages for those with 8 years of education and for those with 16 years of education (college graduates) who: i. Are overweight or obese ii. Have at least one alcoholic drink per month iii. Smoke b. For each category in part (a), calculate the average rate of change with respect to years of education. What do they all have in common? c. What does the graph suggest about the link between education and health? Could there be other factors at play?

Read the Anthology article Slopes on the course website and then describe two practical applications of slopes, one of which is from your own experience.

Examine the following graph of 30-year fixed mortgage rates. Assuming the end date is at the beginning of May, estimate an earlier beginning date (different in each part (a)(c)) to make the case that: a. Mortgage rates have dramatically declined. b. Mortgage rates have exploded. c. Mortgage rates have stayed the same.

Compare the accompanying graphs. a. Which line appears to have the steeper slope? b. Use the intercepts to calculate the slope. Which graph actually does have the steeper slope?

Compare the accompanying graphs. a. Which line appears to have the steeper slope? b. Use the intercepts to calculate the slope. Which graph has the more negative (and hence steeper) slope?

The following graphs show the same function graphed on different scales. a. In which graph does Q appear to be growing at a faster rate? b. In which graph does Q appear to be growing at a near zero rate? c. Explain why the graphs give different impressions.

The following graph shows the National Assessment of Educational Progress (NAEP) math scores for California students, grades 4 and 8, as well as national math scores. a. Summarize the message this graph conveys concerning achievement in math scores. b. What strategy is used to make the increase in math scores seem modest? How could you make the scores seem even more modest? c. What strategy could you use to make the increase more dramatic?

Generate two graphs and on each draw a line through the points (0, 3) and (4, 6), choosing x and y scales such that: a. The first line appears to have a slope of almost zero. b. The second line appears to have a very large positive slope

a. Generate a graph of a line through the points (0, 2) and (5, 2). b. On a new grid, choose different scales so that the line through the same points appears to have a large positive slope. c. What have you discovered?

What are three adjectives (like explosive) that would imply rapid growth?

What are three adjectives (like severe) that would imply rapid decline?

(Graphing program optional.) Examine the data given on women in the U.S. military forces. a. Make the case with graphs and numbers that women are a growing presence in the U.S. military. b. Make the case with graphs and numbers that women are a declining presence in the U.S. military. c. Write a paragraph that gives a balanced picture of the changing presence of women in the military using appropriate statistics to make your points. What additional data would be helpful?

On the following page there is a graph on AIDS in America similar to one that appeared in Chapter 1. The left-hand axis shows both the number of new AIDS cases and deaths from AIDS. In this new version, the right-hand axis shows the number of people living with AIDS. a. Find something encouraging to say about these data by using numerical evidence, including estimated average rates of change. b. Find numerical support for something discouraging to say about the data. c. How might we explain the enormous increase in new AIDS cases reported from 1985 to a peak in 19921993, and the drop-off the following year?

(Course software required.) Open L6: Changing Axis Scales in Linear Functions. Generate a line in the upper left-hand box. The same line will appear graphed in the three other boxes but with the axes scaled differently. Describe how the axes are rescaled in order to create such different impressions.

The accompanying graphs show the same data on the median income of a household headed by a black person as a percentage of the median income of a household headed by a white person. Describe the impression each graph gives and how that was achieved. (Data from the U.S. Bureau of the Census.)

The accompanying graph shows the number of Nobel Prizes awarded in science for various countries between 1901 and 1974. It contains accurate information but gives the impression that the number of prize winners declined drastically in the 1970s, which was not the case. What flaw in the construction of the graph leads to this impression?

The following graph shows the trend in National Assessment of Educational Progress (NAEP) math scores of 17-year-old students by gender from 1973 to 2008. a. Summarize the message this graph conveys concerning achievement in math scores by gender. b. What strategy could be used to make the gap between genders seem greater?

Consider the equation . a. Find the value of A for n 5 0, 1, 20. b. Express your answers to part (a) as points with coordinates (n, A).

Consider the equation B 5 450 1 40n . a. Find the value of B for n 5 0, 1, 20. b. Express your answers to part (a) as points with coordinates (n, B).

Determine if any of the following points satisfy one or both of the equations in Exercises 1 and 2. a. (300, 0) b. (10, 850) c. (15, 1050)

Suppose during a 5-year period the profit P(Y ) (in billions of dollars) for a large corporation was given by , where Y represents the year. a. Fill in the chart. b. What are the units of P(Y )? c. What does the 2 in the equation represent, and what are its units? d. What was the initial profit?

Consider the equation D 5 3.40 1 0.11n a. Find the values of D for n 5 0, 1, 2, 3, 4. b. If D represents the average consumer debt, in thousands of dollars, over n years, what does 0.11 represent? What are its units? c. What does 3.40 represent? What are its units?

Suppose the equations and G(n) 5 12,000 1 800n give the total cost of operating an electrical (E) versus a gas (G) heating/cooling system in a home for n years. a. Find the cost of heating a home using electricity for 10 years. b. Find the cost of heating a home using gas for 10 years. c. Find the initial (or installation) cost for each system. d. Determine how many years it will take before $40,000 has been spent in heating/cooling a home that uses: i. Electricity ii. Gas

If the equation gives the total cost of heating/cooling a home after n years, rewrite the equation using only units of measure

Over a 5-month period at Acadia National Park in Maine, the mean night temperature increased on average 5 degrees Fahrenheit per month. If the initial temperature is 25 degrees Fahrenheit, create a function for the night temperature N(t) for month t, where .

A residential customer in the Midwest purchases gas from a utility company that charges according to the formula C(g) 5 11 1 10.50(g), where C(g) is the cost, in dollars, for g thousand cubic feet of gas. a. Find C(0), C(5), and C(10). b. What is the cost if the customer uses no gas? c. What is the rate per thousand cubic feet charged for using the gas? d. How much would it cost if the customer uses 96 thousand cubic feet of gas (the amount an average Midwest household consumes during the winter months)?

Create the formula for converting degrees centigrade, C, to degrees Fahrenheit, F, if for every increase of 5 degrees centigrade the Fahrenheit temperature increases by 9 degrees, with an initial point of (C, F) 5 (0, 32).

Determine the vertical intercept and the rate of change for each of these formulas: a. P 5 4s c. b. d.

12. A hiker can walk 2 miles in 45 minutes. a. What is his average speed in miles per hour? b. What formula can be used to find the distance traveled, d, in miles in t hours? C 5 5 9 C 5 pd F 2 17.78 C 5 2pr

The cost C(x) of producing x items is determined by adding fixed costs to the product of the cost per item and the number of items produced, x. Below are three possible cost functions C(x), measured in dollars. Match each description in parts (e) 2 (g) with the most likely cost function and explain your choices. a. C(x) 5 125,000 1 75x b. C(x) 5 400,000 1 0.30x c. C(x) 5 250,000 1 800x e. The cost of producing a computer f. The cost of producing a college algebra text g. The cost of producing a CD

A new $25,000 car depreciates in value by $5000 per year. Construct a linear function for the value V of the car after t years.

The state of Pennsylvania has a 6% sales tax on taxable items. (Note: Clothes, food, and certain pharmaceuticals are not taxed in Pennsylvania.) a. Create a formula for the total cost (in dollars) of an item C(p) with a price tag p. (Be sure to include the sales tax.) b. Find C(9.50), C(115.25), and C(1899). What are the units?

a. If S(x) 5 20,000 1 1000x describes the annual salary in dollars for a person who has worked for x years for the Acme Corporation, what is the unit of measure for 20,000? For 1000? b. Rewrite S(x) as an equation using only units of measure. c. Evaluate S(x) for x values of 0, 5, and 10 years. d. How many years will it take for a person to earn an annual salary of $43,000?

The following represent linear equations written using only units of measure. In each case supply the missing unit. a. b. c.

18. Identify the slopes and the vertical intercepts of the lines with the given equations. a. y 5 3 1 5x d. Q 5 35t 2 10 b. (t) 5 2t e. (E) 5 10,000 1 3000E c. y 5 4 calories 5 calories 1 (?) ? (grams of fat) miles 5 miles 1 (?) ? (gallons) inches 5 inches 1 (inches/hour) ? (?)

For each of the following, find the slope and the vertical intercept, then sketch the graph. (Hint: Find two points on the line.) a. y 5 0.4x 2 20 b. P 5 4000 2 200C

Construct an equation and sketch the graph of its line with the given slope, m, and vertical intercept, b. (Hint: Find two points on the line.) a. m 5 2, b 5 23 c. m 5 0, b 5 50 b.

In Exercises 21 to 23 find an equation, generate a small table of solutions, and sketch the graph of the line with the indicated attributes. A line that has a vertical intercept of 22 and a slope of 3.

In Exercises 21 to 23 find an equation, generate a small table of solutions, and sketch the graph of the line with the indicated attributes. A line that crosses the vertical axis at 3.0 and has a rate of change of 22.5.

In Exercises 21 to 23 find an equation, generate a small table of solutions, and sketch the graph of the line with the indicated attributes. A line that has a vertical intercept of 1.5 and a slope of 0.

Estimate b (the y-intercept) and m (the slope) for each of the accompanying graphs. Then, for each graph, write the corresponding linear function.

Assuming m is the slope, identify the graph(s) where: a. m 5 3 b. c. m 5 0 d. m 5 23

Which line(s), if any, have a slope m such that: a. m k 5 2? b. m 5 1 2?

For each set of conditions, construct a linear equation and draw its graph. a. A slope of zero and a y-intercept of 23 b. A positive slope and a vertical intercept of 23 c. A slope of 23 and a vertical intercept that is positive

Construct a linear equation for each of the following conditions. a. A negative slope and a positive y-intercept b. A positive slope and a vertical intercept of 210.3 c. A constant rate of change of $1300/year

Match the graph with the correct equation. a. y 5 x b. y 5 2x c. d. y 5 4x 6.

Match the function with its graph. (Note: There is one graph that has no match.) a. f(x) 5 24 1 3x b. g(x) 5 23x 1 4 c. h(x) 5 4 1 3x

In each part construct three different linear equations that all have: a. The same slope b. The same vertical intercept

Which equation has the steepest slope? a. y 5 2 2 7x b. y 5 2x 1 7 c. y 5 22 1 7x

Given the function Q(t) 5 13 2 5t, construct a related function whose graph: a. Lies five units above the graph of Q(t) b. Lies three units below the graph of Q(t) c. Has the same vertical intercept as Q(t) d. Has the same slope as Q(t) e. Has the same steepness as Q(t) but the slope is positive

Given the equation C(n) 5 30 1 15n, construct a related equation whose graph: a. Is steeper b. Is flatter (less steep) c. Has the same steepness, but the slope is negative

On the same grid, graph (and label with the correct equation) three lines that go through the point (0, 2) and have the following slopes: a. b. m 5 2 c.

On the same grid, graph (and label with the correct equation) three lines that go through the point (0, 2) and have the following slopes: a. b. c. C A B y x m 5 25 6 m 5 22 m 5 21 2

Given the following graphs of three straight lines: a. Which has the steepest slope? b. Which has the flattest slope? c. If the slopes of the lines A, B, and C are m1, m2, and m3, respectively, list them in increasing numerical order. d. In this example, why does the line with the steepest slope have the smallest numerical value?

Write an equation for the line through (22, 3) that has slope: a. 5 b. c. 0

Write an equation for the line through (0, 50) that has slope: a. 220 b. 5.1 c. 0

Calculate the slope and write an equation for the linear function represented by each of the given tables. a. b. A W 5 12 7 16 x y 2 7.6 4 5.1

Determine which of the following tables represents a linear function. If it is linear, write the equation for the linear function.

Plot each pair of points, then determine the equation of the line that goes through the points. a. (2, 3), (4, 0) c. (2, 0), (0, 2) b. (22, 3), (2, 1) d. (4, 2), (25, 2)

Find the equation for each of the lines AC on the accompanying graph.

Put the following equations in y b mx form, then identify the slope and the vertical intercept. a. 2x 2 3y 5 6 b. 3x 1 2y 5 6 c. 1 3 x 1 1 2 y 5 6 d. 2y 2 3x 5 0 e. 6y 2 9x 5 0 f. 1 2 x 2 2 3 y 5 21 6

(Graphing program optional.) Solve each equation for y in terms of x, then identify the slope and the y-intercept. Graph each line by hand. Verify your answers with a graphing utility if available. a. 24y 2 x 2 8 5 0 c. 24x 2 3y 5 9 b. d. 6x 2 5y 5 15

Complete the table for each of the linear functions, and then sketch a graph of each function. Make sure to choose an appropriate scale and label the axes. a. b. x h(x) 5 50x 100 20.5 0 0.5 x (x) 5 0.10x 1 10 2100 0 100

(Graphing program optional.) The equation K 5 4F 2 160 models the relationship between F, the temperature in degrees Fahrenheit, and K, the number of chirps per minute for the snow tree cricket. a. Assuming F is the independent variable and K is the dependent variable, identify the slope and vertical intercept in the given equation. b. Identify the units for K, 4, F, and 2160. c. What is a reasonable domain for this model? d. Generate a small table of points that satisfy the equation. Be sure to choose realistic values for F from the domain of your model. e. Calculate the slope directly from two data points. Is this value what you expected? Why? f. Graph the equation, indicating the domain

Find an equation to represent the cost of attending college classes if application and registration fees are $150 and classes cost $250 per credit.

a. Write an equation that describes the total cost to produce x items if the startup cost is $200,000 and the production cost per item is $15. b. Why is the total average cost per item less if the item is produced in large quantities?

Your bank charges you a $2.50 monthly maintenance fee on your checking account and an additional $0.10 for each check you cash. Write an equation to describe your monthly checking account costs.

If a town starts with a population of 63,500 that declines by 700 people each year, construct an equation to model its population size over time. How long would it take for the population to drop to 53,000?

A teachers union has negotiated a uniform salary increase for each year of service up to 20 years. If a teacher started at $26,000 and 4 years later had a salary of $32,000: a. What was the annual increase? b. What function would describe the teachers salary over time? c. What would be the domain for the function?

Your favorite aunt put money in a savings account for you. The account earns simple interest; that is, it increases by a fixed amount each year. After 2 years your account has $8250 in it and after 5 years it has $9375. a. Construct an equation to model the amount of money in your account. b. How much did your aunt put in initially? c. How much will your account have after 10 years?

You read in the newspaper that the river is polluted with 285 parts per million (ppm) of a toxic substance, and local officials estimate they can reduce the pollution by 15 ppm each year. a. Derive an equation that represents the amount of pollution, P, as a function of time, t. b. The article states the river will not be safe for swimming until pollution is reduced to 40 ppm. If the cleanup proceeds as estimated, in how many years will it be safe to swim in the river?

The womens recommended weight formula from Harvard Pilgrim Healthcare says, Give yourself 100 lb for the first 5 ft plus 5 lb for every inch over 5 ft tall. a. Find a mathematical model for this relationship. Be sure you clearly identify your variables. b. Specify a reasonable domain and range for the function and then graph the function. c. Use your model to calculate the recommended weight for a woman 5 feet, 4 inches tall; and for one 5 feet, 8 inches tall.

In 1977 a math professor bought her condominium in Cambridge, Massachusetts, for $70,000. The value of the condo has risen steadily so that in 2007 real estate agents tell her the condo is now worth $850,000. a. Find a formula in point-slope form, to represent these facts about the value of the condo V(t), as a function of time, t, in years. b. If she retires in 2010, what does your formula predict her condo will be worth then?

The y-axis, the x-axis, the line x 5 6, and the line y 5 12 determine the four sides of a 6-by-12 rectangle in the first quadrant (where x . 0 and y . 0) of the xy plane. Imagine that this rectangle is a pool table. There are pockets at the four corners and at the points (0, 6) and (6, 6) in the middle of each of the longer sides. When a ball bounces off one of the sides of the table, it obeys the pool rule: The slope of the path after the bounce is the negative of the slope before the bounce. (Hint: It helps to sketch the pool table on a piece of graph paper first.) a. Your pool ball is at (3, 8). You hit it toward the y-axis, along the line with slope 2. i. Where does it hit the y-axis? ii. If the ball is hit hard enough, where does it hit the side of the table next? And after that? And after that? iii. Show that the ball ultimately returns to (3, 8). Would it do this if the slope had been different from 2? What is special about the slope 2 for this table? b. A ball at (3, 8) is hit toward the y-axis and bounces off it at . Does it end up in one of the pockets? If so, what are the coordinates of that pocket? c. Your pool ball is at (2, 9). You want to shoot it into the pocket at (6, 0). Unfortunately, there is another ball at (4, 4.5) that may be in the way. i. Can you shoot directly into the pocket at (6, 0)? ii. You want to get around the other ball by bouncing yours off the y-axis. If you hit the y-axis at (0, 7), do you end up in the pocket? Where do you hit the line x 5 6? iii. If bouncing off the y-axis at (0, 7) didnt work, perhaps there is some point (0, b) on the y-axis from which the ball would bounce into the pocket at (6, 0). Try to find that point.

. Find the equation of the line shown on the accompanying graph. Use this equation to create two new graphs, taking care to label the scales on your new axes. For one of your graphs, choose scales that make the line appear steeper than in the original graph. For your second graph, choose scales that make the line appear less steep than in the original graph.

You are living in Paris but will be traveling in London. The exchange rate that a French bank gave in January 2010 was 1 euro (standard European currency) for 0.87 pound (currency in Great Britain). The bank charges 3 euros for any transaction. a. Write a general equation for changing euros into pounds (including transaction fees). b. Graph pounds as a function of euros. c. Would it make any sense to exchange 10 euros?

According to Moodys www.econ.com. in the fourth quarter of 2006, 6.5% of homeowners had negative equity (that is, the market value of the home falls below the outstanding debt to pay off the mortgage). The percentage increased at a roughly constant rate until the fourth quarter of 2008, gaining about 2.5 percentage points per quarter. a. Construct an equation describing the relationship of the percent of homeowners having negative equity vs. the number of quarters since the fourth quarter of 2006. b. Determine a reasonable domain and then graph the function. c. Given this model, what percentage of homeowners had negative equity in the fourth quarter of 2008?

a. Create a third column in Tables A and B, and insert values for the average rate of change. (The first entry will be n.a..) b. In either table, is d a linear function of t? If so, construct a linear equation relating d and t for that table.

Adding minerals or organic compounds to water lowers its freezing point. Antifreeze for car radiators contains glycol (an organic compound) for this purpose. The accompanying table shows the effect of salinity (dissolved salts) on the freezing point of water. Salinity is measured in the number of grams of salts dissolved in 1000 grams of water. So our units for salinity are in parts per thousand, abbreviated ppt. Is the relationship between the freezing point and salinity linear? If so, construct an equation that models the relationship. If not, explain why.

The accompanying data show rounded mean values for blood alcohol concentration (BAC) for people of different weights, according to how many drinks (5 oz wine, 1.25 oz 80-proof liquor, or 12 oz beer) they have consumed. a. Examine the data on BAC for a 100-pound person. Are the data linear? If so, find a formula to express blood alcohol concentration, A, as a function of the number of drinks, D, for a 100-pound person. Blood Alcohol Concentration for Selected Weights Number of Drinks 100 lb 140 lb 180 lb 2 0.075 0.054 0.042 4 0.150 0.107 0.083 6 0.225 0.161 0.125 8 0.300 0.214 0.167 10 0.375 0.268 0.208 b. Examine the data on BAC for a 140-pound person. Are the data linear? If theyre not precisely linear, what might be a reasonable estimate for the average rate of change of blood alcohol concentration, A, with respect to number of drinks, D? Find a formula to estimate blood alcohol concentration, A, as a function of number of drinks, D, for a 140-pound person. Can you make any general conclusions about BAC as a function of number of drinks for all of the weight categories? c. Examine the data on BAC for people who consume two drinks. Are the data linear? If so, find a formula to express blood alcohol concentration, A, as a function of weight, W, for people who consume two drinks. Can you make any general conclusions about BAC as a function of weight for any particular number of drinks?

Using the general formula y mx that describes direct proportionality, find the value of m if: a. y is directly proportional to x and y 2 when x 10. b. y is directly proportional to x and y 0.1 when x 0.2. c. y is directly proportional to x and y 1 when . 2. Fo

For each part, construct an equation and then use it to solve the problem. a. Pressure P is directly proportional to temperature T, and P is 20 lb per square inch when T is 60 degrees Kelvin. What is the pressure when the temperature is 80 degrees Kelvin? b. Earnings E are directly proportional to the time T worked, and E is $46 when T is 2 hours. How long has a person worked if she earned $471.50? c. The number of centimeters of water depth W produced by melting snow is directly proportional to the number of centimeters of snow depth S. If W is 15.9 cm when S is 150 cm, then how many centimeters of water depth are produced by a 100-cm depth of melting snow?

In the accompanying table y is directly proportional to x. x 5 1 4 a. Find the formula relating y and x, then determine the missing value in the table. b. Interpret the coefficient of x in this situation.

The electrical resistance R (in ohms) of a wire is directly proportional to its length l (in feet). a. If 250 feet of wire has a resistance of 1.2 ohms, find the resistance for 150 ft of wire. b. Interpret the coefficient of l in this context

For each of the following linear functions, determine the independent and dependent variables and then construct an equation for each function. a. Sales tax is 6.5% of the purchase price. b. The height of a tree is directly proportional to the amount of sunlight it receives. c. The average salary for full-time employees of American domestic industries has been growing at an annual rate of $1300/year since 1985, when the average salary was $25,000.

On the scale of a map 1 inch represents a distance of 35 miles. a. What is the distance between two places that are 4.5 inches apart on the map? b. Construct an equation that converts inches on the map to miles in the real world.

Find a function that represents the relationship between distance, d, and time, t, of a moving object using the data in the table at the top of the next column. Is d directly proportional to t? Which is a more likely choice for the object, a person jogging or a moving car?

Determine which (if any) of the following variables (w, y, or z) is directly proportional to x:

Find the slope of the line through the pair of points, then determine the equation. a. (2, 3) and (5, 3) c. (3, 8) and (3, 4) b. (4, 7) and (12, 7) d. (2, 3) and (2, 1) 10. D

Describe the graphs of the following equations. a. y 5 22 c. e. y 5 324 b. x 5 22 d. f.

The accompanying figure shows the quantity of books (in millions) shipped by publishers in the United States between 2007 and 2010. Construct the equation of a horizontal line that would be a reasonable model for these data. y 5 2 3

An employee for an aeronautical corporation had a starting salary of $25,000/year. After working there for 10 years and not receiving any raises, he decides to seek employment elsewhere. Graph the employees salary as a function of time for the time he was employed with this corporation. What is the domain? What is the range?

For each of the given points write equations for three lines that all pass through the point such that one of the three lines is horizontal, one is vertical, and one has slope 2. a. (1, 4) b. (2, 0)

Consider the function (x) 4. a. What is (0)? (30)? (12.6)? b. Describe the graph of this function. c. Describe the slope of this functions graph.

A football player who weighs 175 pounds is instructed at the end of spring training that he has to put on 30 pounds before reporting for fall training. a. If fall training begins 3 months later, at what (monthly) rate must he gain weight? b. Suppose that he eats a lot and takes several nutritional supplements to gain weight, but due to his metabolism he still weighs 175 pounds throughout the summer and at the beginning of fall training. Sketch a graph of his weight versus time for those 3 months.

. a. Write an equation for the line parallel to y 2 4x that passes through the point (3, 7). b. Find an equation for the line perpendicular to y 2 4x that passes through the point (3, 7).

a. Write an equation for the line parallel to y 4 x that passes through the point (3, 7). b. Find an equation for the line perpendicular to y 4 x that passes through the point (3, 7). 18

Construct the equation of a line that goes through the origin and is parallel to the graph of the given equation. a. y 5 6 b. x 5 23 c. y 5 2x 1 3

Construct the equation of a line that goes through the origin and is perpendicular to the given equation. a. y 6 b. x 3 c. y x 3 20.

Which lines are parallel to each other? Which lines are perpendicular to each other? a. c. y 5 22x 1 10 e. 2y 1 4x 5 212 b. y 5 3x 2 4 d. y 5 23x 2 2 f. y 2 3x 5 7

Because different scales may be used on the horizontal and vertical axes, it is often difficult to tell if two lines are perpendicular to each other. In parts (a) and (b), determine the equations of each pair of lines and show whether or not the paired lines are perpendicular to each other.

In each part construct the equations of two lines that: a. Are parallel to each other b. Intersect at the same point on the y-axis c. Both go through the origin d. Are perpendicular to each other

For each of the accompanying graphs you dont need to do any calculations or determine the actual equations. Using just the graphs, determine if the slopes for each pair of lines are the same. Are the slopes both positive or both negative, or is one negative and one positive? Do the lines have the same y-intercept?

Find the equation of the line in the form y mx b for each of the following sets of conditions. Show your work. a. Slope is $1400/year and line passes through the point (10 yr, $12,000). b. Line is parallel to 2y 7x y 4 and passes through the point (1, 2). c. Equation is 1.48x 2.00y 4.36 0. d. Line is horizontal and passes through (1.0, 7.2). e. Line is vertical and passes through (275, 1029). f. Line is perpendicular to y 2x 7 and passes through (5, 2). 25. In

In the equation Ax By C: a. Solve for y so as to rewrite the equation in the form y mx b. b. Identify the slope. c. What is the slope of any line parallel to Ax By C? d. What is the slope of any line perpendicular to Ax By C? 26

Use the results of Exercise 25, parts (c) and (d), to find the slope of any line that is parallel and then one that is perpendicular to the given lines. 16 a. 5x 1 8y 5 37 b. 7x 1 16y 5 214 c. 30x 1 47y 5 0

Construct the graphs of the following piecewise linear functions. Be sure to indicate whether an endpoint is included in or excluded from the graph. a. b. y 6543210 0 1 2 3 4 5 6 x g(x) 5 e x 1 3 for 24 , x # 0 2 2 x for 0 , x # 4 f(x) 5 e 2 for 23 , x # 0 1 for 0 , x # 3

Construct piecewise linear functions for the following graphs and determine the domain and range. a. Graph of f(x) b. Graph of g(x)

Given the following graph of g(x) on the next page: a. Construct a piecewise linear function of g(x). b. Construct another function using absolute values. c. Describe the relationship between this graph and the graph of f(x) uxu.

a. Normal human body temperature is often cited as 98.6 F. However, any temperature that is within 1 F more or less than that is still considered normal. Construct an absolute value inequality that describes normal body temperatures T that lie within that range. Then rewrite the expression without the absolute value sign. b. The speed limit is set at 65 mph on a highway, but police do not normally ticket you if you go less than 5 miles above or below that limit. Construct an absolute value inequality that describes the speeds S at which you can safely travel without getting a ticket. Rewrite the expression without using the absolute value sign.

Assume two individuals, A and B, are traveling by car and initially are 400 miles apart on a highway. They travel toward each other, pass and then continue on. a. If A is traveling at 60 miles per hour, and B is traveling at 40 mph, write two functions, dA(t) and dB(t), that describe the distance (in miles) that A and B each has traveled over time t (in hours). b. Now construct a function for the distance DAB (t) between A and B at time t (in hours). Graph the function for 0 t 8 hours. c. At what time will A and B cross paths? At that point, how many miles has each traveled? d. What is the distance between them one hour before they meet? An hour after they meet? Interpret both values in context. (Hint: If they are traveling toward each other, the distance between them is considered positive. Once they have met and are traveling away from each other, the distance between them is considered negative.) e. Now construct an absolute value function that describes the (positive) distance between A and B at any point, and graph your result for 0 t 8 hours.

On Christmas Day 2009, there was a bomb scare aboard a Detroit-bound plane. Soon afterwards, a Gallup poll indicated that 39% of Americans were worried that they or a close friend or family member would become a victim of terrorism. The poll had a 4% margin of error. Create an absolute value inequality that describes the ranges of possible poll percentages and display it on a number line.

The greatest integer function y [x] is defined as the greatest integer x (i.e., it rounds x down to the nearest integer at or below x). a. What is [2] ? [2.5] ? [2.9999999] ? b. Sketch a graph of the greatest integer function for 0 x 5. Be sure to indicate whether each endpoint is included or excluded. [Note: A bank employee embezzled hundreds of thousands of dollars by inserting software to round down transactions (such as generating interest on an account) to the nearest cent, and siphoning the round-off differences into his account. He was eventually caught.]

The following table shows U.S. first-class stamp prices (per ounce) over time. a. Construct a step function describing stamp prices for 20062010. b. Graph the function. Be sure to specify whether each of the endpoints is included or excluded. c. In 2008 the post office created forever stamps that (if you bought them then, at 42 cents) would always be valid. How would this alter the graph if you used only forever stamps?

Sketch a graph for each of the following situations. a. The amount in your savings account over a month, where you direct-deposit your paycheck each week and make one withdrawal during the month. b. The amount of money in an ATM machine over one day, where the ATM is stocked with dollars at the beginning of the day, and then ATM withdrawals of various sizes are made.

A department store is offering a coupon for 20% off for any purchases over $100. a. How much would you spend for a $100 purchase? b. How much would you actually spend if the original purchase was $100.01? $101.00? $111.00? c. Construct a piecewise function for the actual cost, C, of the purchase as a function of the original purchase price, P

A name-brand cookie mix calls for baking cookies at a temperature of 350 F. However, for high altitudes (3500 6500 ft above sea level), baking at 375 F is suggested. a. Construct a piecewise linear function for the recommended baking temperature, T(s), as a function of feet above sea level, s. b. What is a reasonable domain for this function? What is the range?

a. Create a piecewise function and sketch the graph of the following scenario where D(t) is the total distance walked (in miles) as a function time t (in minutes): You begin a walk from your home at a fast pace of 2 miles per hour for 30 minutes. You rest for 15 minutes, and then you continue walking at a leisurely pace of 1 mile per hour for 30 minutes. b. How far have you walked at the end of 30, 45, and 60 minutes?

Create a piecewise linear function and sketch a graph for this scenario, where body temperature, T(x), is a function of time x in hours since 6 a.m. a. You are feeling fine at 6 a.m. and your temperature remains a normal 98.6 F. But by 8 a.m. you start to feel feverish. Your temperature begins to rise at a rate of 0.5 per hour for 4 hours. Your temperature remains at this level for 6 more hours and then gradually decreases at a rate of 0.25 per hour until it is back to normal. b. What is your temperature at 8 a.m., noon, 6 p.m., and 10 p.m.?

To motivate customers, Puget Sound Energy explored ways to set lower electric rates for off-peak energy use and higher rates for hours of peak consumption. Use the chart above to create a piecewise function where E(h) describes the rates per kilowatt hour charged (written on each bar) and h number of hours since 5 a.m. (Note: Use the times 6 a.m., 10 a.m., 5 p.m., and 9 p.m., to translate into values for h.) The jagged lines behind the bars indicate the actual use of electricity.

Interval training is a series of repetitions of work with a recovery period following each repetition. To prepare for a long-distance race such as a marathon, a typical interval training program might consist of running 1 mile at a fast rate, followed by a 3-minute recovery period of jogging. The runner repeats this 5 times. A particular runners training program is to run a mile at an average of 8 mph (or 1/8 0.125 hr/mile), then jog at an average speed of 4 mph (or 1/4 0.25 hr/mile). Create a piecewise function describing her total time, T(d), for one repetition as a function of distance d traveled. Graph the function.

. A 60-year-old U.S. Masters swimmer swam a 200-yard freestyle in 2 minutes. Each lap of the pool is 50 yards. The following table gives her splits (time to swim that 50-yard lap). Assume she swam at a constant rate during a given lap. Create a piecewise linear function S(y) for cumulative seconds as a function of distance, y, in yards.

Match each equation with the appropriate table. a. y 5 3x 1 2 b. c. y 5 1.5x 1 2

Match each of the equations with the appropriate graph. a. y 5 10 2 2x b. y 5 10 2 5x c. y 5 10 2 0.5x

(Graphing program recommended.) Identify which of the following data tables represent exact and which approximate linear relationships. For the one(s) that are exactly linear, construct the corresponding equation(s). For the one(s) that are approximately linear, generate the equation of a best-fit line; that is, plot the points, draw in a line approximating the data, pick two points on the line (not necessarily from your data) to generate the slope, and then construct the equation.

(Graphing program recommended.) Plot the data in each of the following data tables. Determine which data are exactly linear and which are approximately linear. For those that are approximately linear, sketch a line that looks like a best fit to the data. In each case generate the equation of a line that you think would best model the data. a. b. The amount of tax owed on a purchase price. c. The number of pounds in a given number of kilograms. Use the linear equations found in parts (a), (b), and (c) to approximate, respectively, the values for: d. The estimated percentage of high school seniors admitting to drug use in 2010. Check your prediction on the Internet. e. The amount of tax owed on $7.79 and $25.75 purchase prices. f. The number of pounds in 15 kg and 150 kg.

Determine which data represent exactly linear and which approximately linear relationships. For the approximately linear data, sketch a line that looks like a best fit to the data. In each case generate the equation of a line that you think would best model the data. a. The number of solar energy units consumed (in quadrillions of British thermal units, called Btus)

In 2002 the United States consumed 617 million gallons of wine and in 2007, about 745 million gallons. a. Assuming the growth was linear, create a function that could model the trend in wine consumption. b. Estimate the amount of wine consumed in 2000. The actual amount was about 568 million gallons worth of wine. How accurate was your approximation? Source: U.S. Department of Agriculture, Economic Research Service, 2009.

The percentage of medical degrees awarded to women in the United States between 1970 and 2008 is shown in the accompanying graph. The data show that since 1970 the percentage of female doctors has been rising. a. Sketch a line that best represents the data points. Use your line to estimate the rate of change of the percentage of M.D. degrees awarded to women. b. If you extrapolate your line, estimate when 100% of doctors degrees will be awarded to women. c. It seems extremely unlikely that 100% of medical degrees will ever be granted to women. Comment on what is likely to happen to the rate of growth of womens degrees in medicine; sketch a likely graph for the continuation of the data into this century. If possible, check on the Internet for the current number of female medical degrees.

The accompanying graph shows the mortality rates (in deaths per 1000) for male and female infants in the United States from 1980 to 2009. a. Are male or female infants more likely to die? Estimate the infant mortality rates for both sexes in 1980 and in 2009. (Be sure to use the correct units.) b. Both female and male infant mortality rates show a roughly linear decline between 1980 and 2000. Draw two best-fit lines (one female, one male) between 1980 and 2000, calculate their slopes, and interpret what each slope means. c. Now draw two best-fit lines (one female, one male) for infant mortality between 2005 and 2009, calculate their slopes, and interpret what each means. d. Describe the overall trends in U.S. infant mortality since 1980.

The accompanying scatter plot shows the relationship between literacy rate (the percentage of the population who can read and write) and infant mortality rate (infant deaths per 1000 live births) for 91 countries. The raw data are contained in the Excel or graph link file NATIONS and are described at the end of the Excel file. (You might wish to identify the outlier, the country with about a 20% literacy rate and a low infant mortality rate of about 40 per 1000 live births.) Construct a linear model. Show all your work and clearly identify the variables and units. Interpret your results.

The accompanying graph shows data for the mens Olympic 16-pound shot put from 1900 to 2008. c. What do you think the graph would look like after 2015? After 2020? Why? 12. The accompanying graph shows the relationship between the age of a woman when she has her first child and her lifetime risk of getting breast cancer relative to a childless woman. a. The shot put results are roughly linear between 1900 and 1988. Sketch a best-fit line for those years. Estimate the coordinates of two points on the line to calculate the slope. Interpret the slope in this context. b. What is happening to the winning shot put results after 1988? Estimate the slope of the best-fit line for the years after 1988. c. Letting x years since 1900, construct a piecewise linear function S(x) that describes the winning Olympic shot put results (in feet thrown) between 1900 and 2008.

Cell phones have become immensely popular, as the following chart shows. a. A linear model seems reasonable between 1990 and 2007. Sketch a best-fit line for those dates, then pick two points on your line and calculate the slope. Interpret the slope in this context. b. Letting x years since 1990, construct a piecewise linear function P(x) that would describe the percentage of U.S. households with cell phones between 1990 and 2007. c. What do you think the graph would look like after 2015? After 2020? Why?

The accompanying graph shows the relationship between the age of a woman when she has her first child and her lifetime risk of getting breast cancer relative to a childless woman. a. The shot put results are roughly linear between 1900 and 1988. Sketch a best-fit line for those years. Estimate the coordinates of two points on the line to calculate the slope. Interpret the slope in this context. b. What is happening to the winning shot put results after 1988? Estimate the slope of the best-fit line for the years after 1988. c. Letting x years since 1900, construct a piecewise linear function S(x) that describes the winning Olympic shot put results (in feet thrown) between 1900 and 2008. 11. Cell phones have become immensely popular, as the following chart shows. a. A linear model seems reasonable between 1990 and 2007. Sketch a best-fit line for those dates, then pick two points on your line and calculate the slope. Interpret the slope in this context. b. Letting x years since 1990, construct a piecewise linear function P(x) that would describe the percentage of U.S. households with cell phones between 1990 and 2007. a. If a woman has her first child at age 18, approximately what is her risk of developing cancer relative to a woman who has never borne a child? b. At roughly what age are the chances the same that a woman will develop breast cancer whether or not she has a child? c. If a first-time mother is beyond the age you specified in part (b), is she more or less likely to develop breast cancer than a childless woman? d. Sketch a line that looks like a best fit to the data, estimate the coordinates of two points on the line, and use them to calculate the slope. e. Interpret the slope in this context. f. Construct a linear model for these data, identifying your independent and dependent variables.

(Graphing program recommended.) The data in the table below show that health care is becoming more expensive and is taking a bigger share of the U.S. gross domestic product (GDP). The GDP is the market value of all goods and services that have been bought for final use. a. Graph health care costs as a percentage of GDP versus year, with time on the horizontal axis. Measure time in years since 1960. Draw a straight line by eye that appears to be the closest fit to the data. Figure out the slope of your line and create a function H(t) for health cares percentage of the GDP as a function of t, years since 1960. b. What does your formula predict for health care as a percentage of GDP for the year 2010? Check your answer on the Internet if you have online access. c. Why do you think the health care cost per person has gone up so much more dramatically than the health care percentage of the GDP?

Hybrid car sales peaked in early 2008, then showed a steady decline in the second half of 2008. The following graph shows the annual sales of hybrids between 2004 and 2009. a. Relabel years as years since 2004. b. Sketch a line corresponding to increasing hybrid sales. Then calculate and interpret its slope in terms of car sales. Create a linear model, specifying its domain. c. Sketch a line corresponding to decreasing hybrid sales. Then calculate and interpret its slope in terms of car sales. Create a linear model, specifying its domain. d. Generate a piecewise linear function for hybrid sales. e. In July 2009 there were 35,429 hybrids sold out of 999,890 total vehicle sales. What percentage of all vehicle sales were hybrids then? f. If you have online access check on the Internet for 2011 hybrid sales to see if they have improved.

(Graphing program recommended.) The Gas Guzzler Tax is imposed on manufacturers on the sale of new-model cars (not minivans, sport utility vehicles, or pickup trucks) whose fuel economy fails to meet certain statutory regulations, to discourage the production of fuel-inefficient vehicles. The tax is collected by the IRS and paid by the manufacturer. The table (top right) shows the amount of tax that the manufacturer must pay for a vehicles miles per gallon fuel efficiency. a. Plot the data, verify that they are roughly linear, and add a line of best fit. b. Choose two points on the line, find the slope, and then form a linear equation with x as the fuel efficiency in mpg and y as the tax in dollars. c. What is the average rate of change of the amount of tax imposed on fuel-inefficient vehicles? Interpret the units.

A veterinarians office displayed the following table comparing dog age (in dog years) to human age (in human years). The chart shows that the relationship is fairly linear. a. Draw a line that looks like a best fit to the data. b. Estimate the coordinates and label two points on the line. Use them to find the slope. Interpret the slope in this context. c. Using H for human age and D for dog age, identify which you are using as the independent and which you are using as the dependent variable. d. Generate the equation of your line. e. Use the linear model to determine the human age of a dog that is 17 dog years old. f. Middle age in humans is 4559 years. Use your model equation to find the corresponding middle age in dog years. g. What is the domain for your model?

In general, heavier cars get lower or worse gas mileage than lighter cars. The graph on the following page shows the mpg (miles per gallon) for cars of given weights (in 1000 pounds). a. A linear model seems reasonable for the mpg vs. weight of cars. Sketch a line corresponding to decreasing mpg. Then calculate and interpret its slope in terms of mpg and weight. b. Letting w weight (in 1000s of pounds), construct a linear function M(w) that would describe the miles per gallon for cars of weights between 2 thousand and 4 thousand pounds. Create a linear model, specifying its domain. c. Use your model to predict the mpg for a car weighing 3 thousand pounds. d. Use your model to predict the weight of a car getting 30 mpg.

The following scatter plot shows the relationship between the average January temperatures and latitude for 30 U.S. cities. a. Sketch a line through the graph of the data that best represents the relationship of the average January temperature and latitude. Does the line seem to be a reasonable model for the data? What is the approximate slope of the line through two points? Show your work. b. Construct a linear function T(d) that describes the average January temperature as a function of d, the latitude. c. Extra: Find the latitude and the average January temperature of your city and plot the point on the graph. How well did your model in part (b) predict your citys average January temperature?

The figures (top left of next page) show regression lines and corresponding correlation coefficients for four different scatter plots. a. Which of the lines describes the strongest linear relationship between the variables? Which of the lines describes the weakest linear relationship? b. Which lines describe a positive relationship? A negative relationship?

The accompanying graph and regression line show median personal total income vs. years of education past grade 8. Median personal total income 5 3230 1 6490 ? yrs. educ. past grade 8 DATA a. What is the slope of the regression line? b. Interpret the slope in this context. c. By what amount does this regression line predict that median personal total income changes for 1 additional year of education? For 10 additional years of education? d. What features of the data are not well described by the regression line?

The following equation represents the best-fit regression line for median personal earnings vs. years of education for the 304 people in the FAM1000 data set who live in the southern region of the United States. S = 15820 + 2960E where S = median personal earnings in the South and E = years of education past grade 8, and cc = 0.51. a. Identify the slope of the regression line, the vertical intercept, and the correlation coefficient. b. What does the slope mean in this context? c. By what amount does this regression line predict that median personal earnings for those who live in the South change for 1 additional year of education? For 10 additional years of education?

The following equation represents a best-fit regression line for median personal total income of white males vs. years of education past grade eight: median personal total incomewhite males 5 . educ. past grade 8 The correlation coefficient is 0.93 and the sample size is 454 white males. a. What is the rate of change of median personal total income with respect to years of education? b. Generate three points that lie on this regression line. Use two of these points to calculate the slope of the regression line. c. How does this slope relate to your answer to part (a)? d. Sketch the graph.

From the FAM1000 data, the best-fit regression line for median personal total income of white females vs. years of education past grade 8 is: 5 The correlation coefficient is 0.60 and the sample size is 356 white females. a. Interpret the number 2310 in this equation. b. Generate a small table with three points that lie on this regression line. Use two of these points to calculate the slope of the regression line. c. How does this slope relate to your answer to part (a)? d. Sketch the graph. e. Describe some differences between median personal total income vs. education for white females and for white males (see Exercise 4). What are some of the limitations of the model in making this comparison?

We can look for correlations for personal earnings with other variables. Here is a scatter plot of mean personal earnings and age. a. Is there a linear correlation between earnings and age? What would you expect the cc to be? b. There does seem to be an underlying pattern in the scatter plot. Describe the pattern in terms of the personal earnings as a person ages.

The term linear regression was coined in 1903 by Karl Pearson as part of his efforts to understand the way physical characteristics are passed from generation to generation. He assembled and graphed measurements of the heights of fathers and their fully grown sons from more than a thousand families. The independent variable, F, was the height of the fathers. The dependent variable, S, was the mean height of the sons who all had fathers with the same height. The best-fit line for the data points had a slope of 0.516, which is much less than 1. If, on average, the sons grew to the same height as their fathers, the slope would equal 1. Tall fathers would have tall sons and short fathers would have equally short sons. Instead, the graph shows that whereas the sons of tall fathers are still tall, they are not (on average) as tall as their fathers. Similarly, the sons of short fathers are not as short as their fathers. Pearson termed this regression; the heights of sons regress back toward the height that is the mean for that population The equation of this regression line is S 33.73 0.516F, where F height of fathers in inches and S mean height of sons in inches. a. Interpret the number 0.516 in this context. b. Use the regression line to predict the mean height of sons whose fathers are 64 inches tall and of those whose fathers are 73 inches tall. c. Predict the height of a son who has the same height as his father. d. If there were over 1000 families, why are there only 17 data points on this graph?

The book Performing ArtsThe Economic Dilemma studied the economics of concerts, operas, and ballets. It included the following scatter plot and corresponding regression line relating attendance per concert to the number of concerts given, for a major orchestra. What do you think were their conclusions?

(Optional use of technology.) The graph at the top of the next page gives the mean annual cost for tuition and fees at public and private 4-year colleges in the United States since 1985. It is clear that the cost of higher education is going up, but public education is still less expensive than private. The graph suggests that costs of both public and private education versus time can be roughly represented as straight lines. a. By hand, sketch two lines that best represent the data. Calculate the rate of change of education cost per year for public and for private education by estimating the coordinates of two points that lie on the line, and then estimating the slope.b. Construct an equation for each of your lines in part (a). (Set 1985 as year 0.) If you are using technology, generate two regression lines from the Excel or graph link data file EDUCOSTS and compare these equations with the ones you constructed. c. If the costs continue to rise at the same rates for both sorts of schools, what would be the respective costs for public and private education in the year 2010? Does this seem plausible to you? Check your answers on the Internet.

Stroke is the third-leading cause of death in the United States, behind heart disease and cancer. The accompanying graph shows the average neuron loss for a typical ischemic stroke. a. Find the slope of the line by estimating the coordinates of two points on the line. Interpret the meaning of the slope in this context. b. Construct an equation for the line, where n number of neurons in billions and d duration of stroke in hours. c. The mean human forebrain has about 22 billion neurons and the mean stroke lasts about 10 hours. Find the percentage of neurons that are lost from a 10-hour stroke.

(Requires technology.) In Section 2.10 we examined the percentage of adult smokers. Here we break out smokers by sexthe percentage of males who smoke and the percentage of females who smoke. If you are male, use only the male data; if you are female, only the female data. a. Construct a scatter plot of the percentage of smokers of your sex. b. On your graph sketch a best-fit line. By estimating coordinates of points on your best-fit line, calculate the slope of the line. What does this tell you? c. Reinitialize your data by setting 1965 as year 0. Now use technology to generate a linear regression line and correlation coefficient (cc) to your new data set. Record the equation and the cc. How good a fit is the regression line to the data? What does the vertical intercept tell you? The slope? d. Write a 60-second summary about the smoking trends for your sex. If possible, compare your results to those of someone of the opposite sex.

(Requires technology.) The accompanying table shows the calories per minute burned by a 154- pound person moving at speeds from 2.5 to 12 miles/hour (mph). (Note: A fast walk is about 5 mph; faster than that is considered jogging or slow running.) Marathons, about 26 miles long, are now run in slightly over 2 hours, so the top distance runners are approaching a speed of 13 mph. a. Plot the data. b. Does it look as if the relationship between speed and calories per minute is linear? If so, generate a linear model. Identify the variables and a reasonable domain for the model, and interpret the slope and vertical intercept. How well does your line fit the data? c. Describe in your own words what the model tells you about the relationship between speed and calories per minute.

(Optional use of technology.) The accompanying graph shows the winning running times in minutes for women in the Boston Marathon a. The graph is roughly linear with negative slope between 1972 and 1982 but seems to flatten out after that. Sketch a piecewise linear function that best approximates the data. b. Set 1972 as year 0 and create a piecewise linear function that describes winning times for women. If using technology, you can create 2 regression lines from 2 subsets of the data, using either the Excel or graph link file MARATHON. c. Interpret the slopes of the 2 linear pieces. d. What would you predict for the winning time in 2010? Since the 2010 Marathon has already occurred, check your prediction on the Internet. e. Write a 60-second summary describing the Boston Marathon winning times for women.

(Optional use of technology.) The accompanying graph shows the world record times for the mens mile. As of January 2010, the 1999 record still stands. Note that several times the standing world record was broken more than once during a year. a. Generate a line that approximates the data (by hand or, if using technology, with the data in the Excel or graph link file MENSMILE. (Set 1910 or 1913 as year 0). If you are using technology, specify the correlation coefficient. Interpret the slope of your line in this context. b. If the world record times continue to change at the rate specified in your linear model, predict the record time for the mens mile in 2010. Check your answer on the Internet. c. In what year would your linear model predict the world record to be 0 minutes? Since this is impossible, what do you think is a reasonable domain for your model? Describe what you think would happen in the years after those included in your domain. d. Write a short paragraph summarizing the trends in the world record times for the mens mile.

The temperature at which water boils is affected by the difference in atmospheric pressure at different altitudes above sea level. The classic cookbook The Joy of Cooking by Irma S. Rombauer and Marion Rombauer Becker gives the data in the accompanying table (rounded to the nearest degree) on the boiling temperature of water at different altitudes above sea level. a. Use the accompanying table for the following: i. Plot boiling temperature in degrees Fahrenheit, F, vs. the altitude. Find a formula to describe the boiling temperature of water, in F, as a function of altitude. ii. According to your formula, what is the temperature at which water will boil where you live? Can you verify this? What other factors could influence the temperature at which water will boil? b. The highest point in the United States is Mount McKinley in Alaska, at 20,320 feet above sea level; the lowest point is Death Valley in California, at 285 feet below sea level. You can think of distances below sea level as negative altitudes from sea level. At what temperature in degrees Fahrenheit will water boil in each of these locations according to your formula? c. Using your formula, find an altitude at which water can be made to boil at 32F, the freezing point of water at sea level. At what altitude would your formula predict this would happen? (Note that airplane cabins are pressurized to near sea-level atmospheric pressure conditions in order to avoid unhealthy conditions resulting from high altitude.)

(Optional use of technology.) The accompanying graph shows the world distance records for the womens long jump. Several times a new long jump record was set more than once during a given year. a. Generate a line that approximates the data by hand or, if using technology, use the data from the Excel or graph link file LONGJUMP. (Set 1950 or 1954 as year 0). Interpret the slope of your line in this context. If you are using technology, specify the correlation coefficient. b. If the world record distances continue to change at the rate described in your linear model, predict the world record distance for the womens long jump in the year 2010. Check on the Internet. c. What would your model predict for the record in 1943? How does this compare with the actual 1943 record of 6.25 meters? What do you think would be a reasonable domain for your model? What do you think the data would look like for years outside your specified domain?

(Optional use of technology.) The accompanying graph shows the increasing number of motor vehicle registrations (cars and trucks) in the United States. a. Using 1945 as the base year, find a linear equation that would be a reasonable model for the data. If using technology, use the data in the Excel or graph link file MOTOR. b. Interpret the slope of your line in this context. c. What would your model predict for the number of motor vehicle registrations in 2007? How does this compare with the actual data value of 237.4 million? d. Using your model, how many motor vehicles will be registered in the United States in 2010? Check your answer on the Internet.

(Requires technology.) Examine the following data on U.S. union membership from 1930 to 2009. a. Using the table, describe the overall trends in the size of the labor force, in the number of union members, and in the percentage of the labor force that belongs to a union. b. Graph the percentage of the labor force in unions vs. time from 1950 to 2009. Measuring time in years since 1950, find a linear regression formula for these data using technology. c. When does your formula predict that only 10% of the labor force will be unionized? d. What data would you want to examine to understand why union membership is declining?

If a study shows that smoking and lung cancer have a high positive correlation, does this mean that smoking causes lung cancer? Explain your answer.

Parental income has been found to have a high positive correlation with their childrens academic success. What are two different ways you could interpret this finding?

According to the National Association of Realtors, the median price for a single-family home fell from $217,900 in 2007 to $173,200 in 2009. Describe the change in the following ways. a. The absolute change in dollars b. The percent decrease c. The annual average rate of change of the price with respect to year (Note: Be sure to include units in all your answers.)

a. According to Apple Computer, sales of its iPod (the worlds best-selling digital audio player) soared from 376,000 units in 2002 to an all-time high of 54,838,000 units in 2008. What was the average rate of change in iPods sold per year over that time period? b. Apples overall sales continue to rise (with iPhone sales), but iPod sales declined to 54,132,000 units in 2009. What was the average rate of change in iPod unit sales between 2008 and 2009?

Given the following graph of the function h(t), identify any interval(s) over which: a. The function is positive; is negative; is zero b. The slope is positive; is negative; is zero

Which line has the steeper slope? Explain why.

You are traveling abroad and realize that American, British, and French clothing sizes are different. The accompanying table shows the correspondence among female clothing sizes in these different countries. a. Are the British and French sizes both linear functions of the U.S. sizes? Why or why not? b. Write a sentence describing the relationship between British sizes and U.S. sizes. c. Construct an equation that describes the relationship between French and U.S. sizes.

A bathtub that initially holds 50 gallons of water starts draining at 10 gallons per minute. a. Construct a function W(t) for the volume of water (in gallons) in the tub after t minutes. b. Graph the function and label the axes. Evaluate W(0) and W(5) and describe what they represent in this context. c. How would the original function and its graph change if the initial volume were 60 gallons? Call the new function U(t). d. How would the original function and graph change if the drain rate were 12 gallons per minute? Call the new function V(t). e. Add the graphs of U(t) and V(t) to the original graph of W(t)

Cell phones have produced a seismic cultural shift. No other recent invention has incited so much praiseand criticism. In 2000 there were 109.4 million cell phone subscriptions in the United States; since then subscriptions steadily increased to reach 262.7 million in 2008. (Note: Some people had more than one subscription.) a. What was the average rate of change in millions of cell phone subscriptions per year between 2000 and 2008? b. Construct a linear function C(t) for cell phone subscriptions (in millions) for t 5 years from 2000. c. If U.S. cell phone subscriptions continue to increase at the same rate, how many will there be in 2015? Does your result sound plausible?

The following graph shows average salaries for major league baseball players from 1990 to 2009. a. On the graph, sketch a straight line that is an approximate mathematical model for the data. b. Specify the coordinates of two points on your line and describe what they represent in terms of your model. c. Calculate the slope and then interpret it in terms of the salary in millions of dollars and the time interval. d. Let S(x) be a linear approximation of the salaries where x 5 years since 2000. What is the slope of the graph of S(x)? The vertical intercept? What is the equation for S(x)? e. Use your model to predict the average salary for 2010. Then, if possible, check it on the Internet.

Match one or more of the following graphs with the following descriptions. Be sure to note the scale on each axis. a. Represents a constant rate of change of 3 b. Has a vertical intercept of 10 c. Is parallel to the line y 5 5 2 2x d. Has a steeper slope than the line C

a. If you travel to China, the exchange rate at the Beijing airport is 1 yuan for 0.147 American dollar and at the hotel is 1 yuan for 0.145 American dollar. Which exchange rate should you use? b. Use the most favorable rate to construct a function Y to convert dollars, d, to yuan, Y(d). c. What is Y(50)? Y(200)?

In May 2010 the Gallup poll reported that 46% of Americans said that they lived with a lot of happiness/enjoyment without a lot of stress/worry. This poll had a 3% margin of error. Create an absolute value inequality that describes the ranges of possible poll percentages and display it on a number line.

Given the following graph: a. Find the equations of line A and line B. b. Show that line A is (or is not) perpendicular to line B.

According to the Environmental Protection Agencys National Coastal Conditions Report II, more than half of the U.S. population live in the narrow coastal fringes. Increasing population in these areas contributes to degradation (by runoff, sewage spills, construction, and overfishing) of the same resources that make the coasts desirable. In 2003 about 153 million people lived in coastal counties. This coastal population is currently increasing by an average of 3600 people per day. a. What is the average rate of change in millions of people per year? b. Let x 5 years since 2003 and construct a linear function P(x) for the coast population (in millions). c. Graph the function P(x) over a reasonable domain. d. What does your model give for the projected population in 2010? Check the actual population on the Internet.

Generate the equation for a line under each of the following conditions. a. The line goes through the points (21, 24) and (4, 6). b. The line is parallel to the line in part (a) and goes through the point (0, 5). c. The line is perpendicular to the line in part (a) and goes through the origin.

In 2005 a record 51.5% of paper consumed in the United States (51.3 million tons) was recycled. The American Forest and Paper Association states that its goal is to have 55% recovery by 2012. a. Plot two data points that represent the percentage of paper recycled at t number of years since 2005. (You may want to crop the vertical axis to start at 50%.) Connect the two points with a line and calculate its slope. b. Assuming linear growth, the line represents a model, R(t), for the percentage of paper recycled at t years since 2005. Find the equation for R(t). c. Find R(0), R(5), and R(20). What does R(20) represent?

Sketch a graph that could represent each of the following situations. Be sure to label your axes. a. The volume of water in a kettle being filled at a constant rate. b. The height of an airplane flying at 30,000 feet for 3 hours. c. The unlimited demand for a certain product. Assume a company produces 10,000 items and consumers will pay virtually any price for one of them. (Use the convention of economists, placing price on the vertical axis and quantity on the horizontal.) d. The price of subway tokens that periodically increase over time. e. The distance from your destination over time when you travel three subway stops to get to your destination.

Generate an equation for each line A, B, C, D, and E shown in the accompanying graph.

U.S. life expectancy has been steadily increasing (see Exercise 15 in Section 2.1 and data file for LIFEPXEC). The following graph shows the roughly linear trends for both males and females over more than three decades. a. According to these data, what is the life expectancy of a female born in 1970? Of a male born in 2005? b. Sketch a straight line to create a linear approximation of the data for your gender. Males and females will have different lines. c. Identify two points on your line and calculate the slope. What does your slope mean in this context? The male slope is steeper than the female slope. What does that imply? d. Generate a linear function for your line. (Hint: Let the independent variable be the number of years since 1970.) e. What would your model predict for your sexs life expectancy in 2010? How close does your model come to Census Bureau current predictions of life expectancies in 2010 of 75.6 years for males and 81.4 for females?

For the function g(x) 5 uxu 1 2: a. Generate a table of values for g(x) using integer values of x between 23 and 3. b. Use the table to sketch a graph of g(x). c. How does this graph compare with the graph of f(x) 5 uxu?

In Exercise 7 in Section 2.9 we met the greatest integer function f(x) 5 [x], where [x] is the greatest integer #x; (i.e., you round down to the nearest integer). So [21.5] 5 22 and [3.99] 5 3. This function is sometimes written using the notation JxK and called the floor function. There is a similar ceiling function, g(x) 5

The table below gives historical data on voter turnout as a percentage of voting-age population (18 years and older) in U.S. presidential elections since 1960. Using selected data from the table, graphs, and dramatically persuasive language, provide convincing arguments in a 60-second summary that during the years 1960 to 2008: a. Voter turnout has plummeted. b. Voter turnout has soared.

a. Construct the equation of a line y1 whose graph is steeper than that of 8x 1 2y 5 6 and does not pass through quadrant I (where both x . 0 and y . 0). Plot the lines for both equations on the same graph. b. Construct another equation of a line y2 that is perpendicular to the original equation, 8x 1 2y 6, and does go through quadrant I. Add the plot of this line to the graph in part (a).

Sweden has kept meticulous records of its population for many years. The following graph shows the mortality rate (as a percentage) for female and male children in Sweden over a 250-year period. a. Describe the overall trend, and the similarities and differences between the female and male graphs. b. Assuming the graphs are roughly linear between 1800 and 1950, draw a line approximating the female childhood mortality rate during this time. What is the slope of the line and what does it tell you in this context? c. Construct a linear function to model the deaths over this period. d. What is happening in the years after 1950? Note: The large spike between 1750 and 1800 represents the more than 300,000 Swedish children (out of a total population of about 2 million) who died from smallpox, the most feared disease of that century

According to the U.S. Environmental Protection Agency, carbon dioxide makes up 84.6% of the total U.S. greenhouse gas emissions. Carbon dioxide, or CO2, arises from the combustion of coal, oil, and gas. Greenhouse gases trap heat on our planet, causing it to become warmer. Effects of this phenomenon are already being seen in increased melting of glaciers and permafrost. If this continues unchecked, water levels will rise, causing flooding in coastal communities, where the majority of the U.S. population lives. (See Exercise 13.) The graph at the top of the next page shows the average rate of change in the concentration of atmospheric CO2 since 1990. (Note that before 1900 the average rate of change in CO2 emissions was typically 0.2 ppm or below.) a. Is the average rate of change always positive? If so, does that imply that the amount of atmospheric carbon dioxide is always increasing? Explain your answer. b. What does it mean when the average rate of change was decreasing (though not negative)?

The following three graphs show U.S. farm income for three crops over the years 20002007. a. Identify each scatter plot as showing a positive, negative, or no discernible correlation. b. If you were a farmer and wanted to boost your farm income, which crop do you think you would plant based on the trend in these graphs? Check the Internet to see if you chose wisely.

The World Health Organization (WHO) states: Access to drinking water and improved sanitation is a fundamental need and a human right vital for the dignity and health of all people. Not surprisingly, there is a correlation between access to water and access to sanitation in the following scatter plot showing data for both urban and rural areas of 185 countries. The computer-generated regression line (from Excel using data file WATERSAN) is and has a cc 0.75. a. Identify the slope and interpret it in context. b. Identify the vertical intercept of the equation. What would be a reasonable domain for the regression line and why? c. How do you know the regression is a reasonable fit to the line? d. Create a newspaper headline to summarize your results.

If f(x) 5 y is decreasing throughout, then the y values decrease as the x values decrease.

Having a slope of zero is the same as having an undefined slope.

A vertical line can be described by a linear function.

Two nonvertical lines that are perpendicular must have slopes of opposite sign

. All linear functions can be written in the form y 5 b 1 mx

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables Linear and decreasing with positive vertical intercept

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Linear and horizontal with vertical intercept 0

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Linear with positive horizontal intercept and negative vertical intercept

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Linear and does not pass through the first quadrant (where x . 0 and y . 0)

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables Linear with average rate of change of 37 minutes/lap

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Linear describing the value of a stock that is currently at $19.25 per share and is increasing exactly $0.25 per quarter

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Two linear functions that are parallel, such that moving one of the functions horizontally to the right two units gives the graph of the other

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Four distinct linear functions all passing through the point (0, 4)

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Two linear functions where the slope of one is m and the slope of the second is , where m is a negative number

In Problems 3241, give an example of a function or functions with the specified properties. Express your answers using equations, and specify the independent and dependent variables. Five data points for which the average rates of change between consecutive points are positive and the sequential points are increasing at a decreasing rate

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample If the average rate of change between any two points of a data set is constant, then the data are linear.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample If the slope of a linear function is negative, then the average rate of change decreases.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample For any two distinct points, there is a linear function whose graph passes through them.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample To write the equation of a specific linear function, one needs to know only the slope.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample Function A in the accompanying figure is increasing at a faster rate than function B.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample The graph of a linear function is a straight line.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample Vertical lines are not linear functions because they cannot be written in the form y 5 b 1 mx, as they have undefined slope.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample There exist linear functions that slant upward moving from left to right but have negative slope.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample A constant average rate of change means that the slope of the graph of a function is zero.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample All linear functions in x and y describe a relationship where y is directly proportional to x.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample The range of the function h(t) 5 is always positive for any value of t.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample The function h(t) 5 can be written as a piecewise linear function.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample A linear model would be appropriate for Barry Bonds graph of home runs vs. number of games.

Is each of the statements in Problems 4255 true or false? If a statement is true, explain how you know. If a statement is false, give a counterexample The graph below shows a step function.