Free-Fall Motion As a promotion for the Houston Astros

In Exercises 1–2, write an equation in slope-intercept form for a line with the given slope m and y-intercept b. m = 8, b = 3.6

In Exercises 1–2, write an equation in slope-intercept form for a line with the given slope m and y-intercept b. m = -1.8, b = -2

In Exercises 3–4, write an equation for the line containing the given points. Graph the line and points. (-2, 4) and (3, 1)

In Exercises 3–4, write an equation for the line containing the given points. Graph the line and points. (1, 5) and (-2, -3)

In Exercises 5-8, expand the expression. \((x+3)^{2}\)

In Exercises 5–8, expand the expression. \((x-4)^{2}\)

In Exercises 5–8, expand the expression. \(3(x-6)^{2}\)

In Exercises 5–8, expand the expression. \(-3(x+7)^{2}\)

In Exercises 9–10, factor the trinomial. \(2 x^{2}-4 x+2\)

In Exercises 9–10, factor the trinomial. \(3 x^{2}+12 x+12\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=3 x^{-5}+17\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=-9+2 x\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=2 x^{5}-\frac{1}{2} x+9\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(f(x)=13\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(h(x)=\sqrt[3]{27 x^{3}+8 x^{6}}\)

In Exercises 1–6, determine which are polynomial functions. For those that are, state the degree and leading coefficient. For those that are not, explain why not. \(k(x)=4 x-5 x^{2}\)

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(-5) = -1 and ƒ(2) = 4

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(-3) = 5 and ƒ(6) = -2

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(-4) = 6 and ƒ(-1) = 2

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(1) = 2 and ƒ(5) = 7

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(0) = 3 and ƒ(3) = 0

In Exercises 7–12, write an equation for the linear function ƒ satisfying the given conditions. Graph y = ƒ(x). ƒ(-4) = 0 and ƒ(0) = 2

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=2(x+1)^{2}-3\)

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=3(x+2)^{2}-7\)

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=4-3(x-1)^{2}\)

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=12-2(x-1)^{2}\)

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=2(x-1)^{2}-3\)

In Exercises 13–18, match a graph to the function. Explain your choice. \(f(x)=12-2(x+1)^{2}\)

In Exercises 19–22, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch each graph by hand. \(g(x)=(x-3)^{2}-2\)

In Exercises 19–22, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch each graph by hand. \(h(x)=\frac{1}{4} x^{2}-1\)

In Exercises 19–22, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch each graph by hand. \(g(x)=\frac{1}{2}(x+2)^{2}-3\)

In Exercises 19–22, describe how to transform the graph of \(f(x)=x^{2}\) into the graph of the given function. Sketch each graph by hand. \(h(x)=-3 x^{2}+2\)

In Exercises 23–26, find the vertex and axis of the graph of the function. \(f(x)=3(x-1)^{2}+5\)

In Exercises 23–26, find the vertex and axis of the graph of the function. \(g(x)=-3(x+2)^{2}-1\)

In Exercises 23–26, find the vertex and axis of the graph of the function. \(f(x)=5(x-1)^{2}-7\)

In Exercises 23–26, find the vertex and axis of the graph of the function. \(g(x)=2(x-\sqrt{3})^{2}+4\)

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(f(x)=3 x^{2}+5 x-4\)

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(f(x)=-2 x^{2}+7 x-3\)

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(f(x)=8 x-x^{2}+3\)

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(f(x)=6-2 x+4 x^{2}\)

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(g(x)=5 x^{2}+4-6 x\)

In Exercises 2732, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form.h1x2 = -2x2- 7x - 4

In Exercises 27–32, find the vertex and axis of the graph of the function. Rewrite the equation for the function in vertex form. \(h(x)=-2 x^{2}-7 x-4\)

In Exercises 33–38, use completing the square to describe the graph of each function. Support your answers graphically \(g(x)=x^{2}-6 x+12\)

In Exercises 33–38, use completing the square to describe the graph of each function. Support your answers graphically \(f(x)=10-16 x-x^{2}\)

In Exercises 33–38, use completing the square to describe the graph of each function. Support your answers graphically \(h(x)=8+2 x-x^{2}\)

In Exercises 33–38, use completing the square to describe the graph of each function. Support your answers graphically \(f(x)=2 x^{2}+6 x+7\)

In Exercises 33–38, use completing the square to describe the graph of each function. Support your answers graphically \(g(x)=5 x^{2}-25 x+12\)

In Exercises 39–42, write an equation for the parabola shown, using the fact that one of the given points is the vertex.

In Exercises 39–42, write an equation for the parabola shown, using the fact that one of the given points is the vertex.

In Exercises 39–42, write an equation for the parabola shown, using the fact that one of the given points is the vertex.

In Exercises 39–42, write an equation for the parabola shown, using the fact that one of the given points is the vertex.

In Exercises 43 and 44, write an equation for the quadratic function whose graph contains the given vertex and point. Vertex (1, 3), point (0, 5)

In Exercises 43 and 44, write an equation for the quadratic function whose graph contains the given vertex and point. Vertex (-2, -5), point (-4, -27)

In Exercises 45–48, describe the strength and direction of the linear correlation

In Exercises 45–48, describe the strength and direction of the linear correlation

In Exercises 45–48, describe the strength and direction of the linear correlation

In Exercises 45–48, describe the strength and direction of the linear correlation

Comparing Age and Weight A group of male children were weighed. Their ages and weights are recorded in Table 2.4. (a) Draw a scatter plot of these data. (b) Writing to Learn Describe the strength and direction of the correlation between age and weight.

Life Expectancy Table 2.5 shows the average number of additional years a U.S. citizen is expected to live for various ages. (a) Draw a scatter plot of these data. (b) Writing to Learn Describe the strength and direction of the correlation between age and life expectancy

Straight-Line Depreciation Mai Lee bought a computer for her home office and depreciated it over 5 years using the straight-line method. If its initial value was $2350, what is its value 3 years later?

Costly Doll Making Patrick’s doll-making business has weekly fixed costs of $350. If the cost for materials is $4.70 per doll and his total weekly costs average $500, about how many dolls does Patrick make each week?

Table 2.6 shows the average hourly compensation of U.S. production workers for several years. Let x be the number of years since 1970, so that x = 5 stands for 1975, and so forth. (a) Writing to Learn Find the linear regression model for the data. What does the slope in the regression model represent? (b) Use the linear regression model to predict the production worker average hourly compensation in the year 2015.

Finding Maximum Area Among all the rectangles whose perimeters are 100 ft, find the dimensions of the one with maximum area.

Determining Revenue The per unit price p (in dollars) of a popular toy when x units (in thousands) are produced is modeled by the function price = p = 12 - 0.025x. The revenue (in thousands of dollars) is the product of the price per unit and the number of units (in thousands) produced. That is, revenue = xp = x(12 - 0.025x). (a) State the dimensions of a viewing window that shows a graph of the revenue model for producing 0 to 100,000 units. (b) How many units should be produced if the total revenue is to be $1,000,000?

Finding the Dimensions of a Painting A large painting in the style of Rubens is 3 ft longer than it is wide. If the wooden frame is 12 in. wide, the area of the picture and frame is \(208\mathrm{\ ft}^2\), find the dimensions of the painting.

Using Algebra in Landscape Design Julie Stone designed a rectangular patio that is 25 ft by 40 ft. This patio is surrounded by a terraced strip of uniform width planted with small trees and shrubs. If the area A of this terraced strip is \(504\mathrm{\ ft}^2\), find the width x of the strip.

Management Planning The Welcome Home apartment rental company has 1600 units available, of which 800 are currently rented at $300 per month. A market survey indicates that each $5 decrease in monthly rent will result in 20 new leases. (a) Determine a function R(x) that models the total rental income realized by Welcome Home, where x is the number of $5 decreases in monthly rent. (b) Find a graph of R(x) for rent levels between $175 and $300 (that is, 0 \leq x \leq 25) that clearly shows a maximum for R(x). (c) What rent will yield Welcome Home the maximum monthly income?

Group Activity Beverage Business The Sweet Drip Beverage Co. sells cans of soda pop in machines. It finds that sales average 26,000 cans per month when the cans sell for 50¢ each. For each nickel increase in the price, the sales per month drop by 1000 cans. (a) Determine a function R(x) that models the total revenue realized by Sweet Drip, where x is the number of $0.05 increases in the price of a can. (b) Find a graph of R(x) that clearly shows a maximum for R(x). (c) How much should Sweet Drip charge per can to realize the maximum revenue? What is the maximum revenue?

Group Activity Sales Manager Planning Jack was named District Manager of the Month at the Athens Wire Co. due to his hiring study. It shows that each of the 30 salespersons he supervises average $50,000 in sales each month, and that for each additional salesperson he would hire, the average sales would decrease $1000 per month. Jack concluded his study by suggesting a number of salespersons that he should hire to maximize sales. What was that number?

Free-Fall Motion As a promotion for the Houston Astros downtown ballpark, a competition is held to see who can throw a baseball the highest from the front row of the upper deck of seats, 83 ft above field level. The winner throws the ball with an initial vertical velocity of 92 ft/sec and it lands on the infield grass. (a) Find the maximum height of the baseball. (b) How much time is the ball in the air? (c) Determine its vertical velocity when it hits the ground.

Baseball Throwing Machine The Sandusky Little League uses a baseball throwing machine to help train 10-year-old players to catch high pop-ups. It throws the baseball straight up with an initial velocity of 48 ft/sec from a height of 3.5 ft. (a) Find an equation that models the height of the ball t seconds after it is thrown. (b) What is the maximum height the baseball will reach? How many seconds will it take to reach that height?

Fireworks Planning At the Bakersville Fourth of July celebration, fireworks are shot by remote control into the air from a pit that is 10 ft below the earth’s surface. (a) Find an equation that models the height of an aerial bomb t seconds after it is shot upward with an initial velocity of 80 ft/sec. Graph the equation. (b) What is the maximum height above ground level that the aerial bomb will reach? How many seconds will it take to reach that height?

Landscape Engineering In her first project after being employed by Land Scapes International, Becky designs a decorative water fountain that will shoot water to a maximum height of 48 ft. What should be the initial velocity of each drop of water to achieve this maximum height? (Hint: Use a grapher and a guess-and-check strategy.)

Patent Applications Using quadratic regression on the data in Table 2.7, predict the year when the number of patent applications reached 450,000. Let x = 0 stand for 1980, x = 10 for 1990, and so forth.

Highway Engineering Interstate 70 west of Denver, Colorado, has a section posted as a 6% grade. This means that for a horizontal change of 100 ft there is a 6-ft vertical change (a) Find the slope of this section of the highway. (b) On a highway with a 6% grade what is the horizontal distance required to climb 250 ft? (c) A sign along the highway says 6% grade for the next 7 mi. Estimate how many feet of vertical change there are along those 7 mi. (There are 5280 ft in 1 mile.)

A group of female children were weighed. Their ages and weights are recorded in Table 2.8. (a) Draw a scatter plot of the data. (b) Find the linear regression model. (c) Interpret the slope of the linear regression equation. (d) Superimpose the regression line on the scatter plot. (e) Use the regression model to predict the weight of a 30-month-old girl.

Table 2.9 shows the median U.S. income of women (in 2007 dollars) for selected years. Let x be the number of years since 1940. (a) Find the linear regression model for the data. (b) Use it to predict the median U.S. female income in 2015.

In Exercises 69–70, complete the analysis for the given Basic Function. Analyzing a Function

In Exercises 69–70, complete the analysis for the given Basic Function. Analyzing a Function

True or False The initial value of \(f(x)=3 x^{2}+2 x-3\) is 0. Justify your answer.

True or False The graph of the function \(f(x)=x^{2}-x+1\) has no x-intercepts. Justify your answer.

In Exercises 73–76, you may use a graphing calculator to solve the problem. In Exercises 73 and 74, \(f(x)=m x+b, f(-2)=3, \text { and } f(4)=1\). Multiple Choice What is the value of m? (A) 3 (B) -3 (C) -1 (D) 1/3 (E) -1/3

In Exercises 73–76, you may use a graphing calculator to solve the problem. In Exercises 73 and 74, \(f(x)=m x+b, f(-2)=3, \text { and } f(4)=1\). Multiple Choice What is the value of b? (A) 4 (B) 11/3 (C) 7/3 (D) 1 (E) -1/3

In Exercises 73–76, you may use a graphing calculator to solve the problem. In Exercises 75 and 76, \(f(x)=2(x+3)^{2}-5\). Multiple Choice What is the axis of symmetry of the graph of ƒ? (A) x = 3 (B) x = -3 (C) y = 5 (D) = y = -5 (E) y = 0

In Exercises 73–76, you may use a graphing calculator to solve the problem. In Exercises 75 and 76, \(f(x)=2(x+3)^{2}-5\). Multiple Choice What is the vertex of ƒ? (A) (0, 02) (B) (3, 5) (C) (3, -5) (D) = (-3, 5) (E) (-3, -5)

Writing to Learn Identifying Graphs of Linear Functions (a) Which of the lines graphed on the next page are graphs of linear functions? Explain. (b) Which of the lines graphed on the next page are graphs of functions? Explain. (c) Which of the lines graphed on the next page are not graphs of functions? Explain.

Average Rate of Change Let \(f(x)=x^2,\ g(x)=3x+2,\ h(x)=7x-3,\ k(x)=mx+b,\text{ and }l(x)=x^3\text{. }\) (a) Compute the average rate of change of ƒ from x = 1 to x = 3. (b) Compute the average rate of change of ƒ from x = 2 to x = 5. (c) Compute the average rate of change of ƒ from x = a to x = c. (d) Compute the average rate of change of g from x = 1 to x = 3. (e) Compute the average rate of change of g from x = 1 to x = 4. (f) Compute the average rate of change of g from x = a to x = c. (g) Compute the average rate of change of h from x = a to x = c. (h) Compute the average rate of change of k from x = a to x = c. (i) Compute the average rate of change of l from x = a to x = c.

Minimizing Sums of Squares The linear regression line is often called the least-square lines because it minimizes the sum of the squares of the residuals, the differences between actual y values and predicted y values: \(\text { residual }=y_{i}-\left(a x_{i}+b\right)\) where \(\left(x_{i}, y_{i}\right)\) are the given data pairs and y = ax + b is the regression equation, as shown in the figure. Use these definitions to explain why the regression line obtained from reversing the ordered pairs in Table 2.2 is not the inverse of the function obtained in Example 3.

Median-Median Line Read about the median-median line by going to the Internet, your grapher owner’s manual, or a library. Then use the following data set to complete this problem \(\{(2,8),(3,6),(5,9),(6,8),(8,11),(10,13),(12,14),(15,4)\}\) (a) Draw a scatter plot of the data. (b) Find the linear regression equation and graph it. (c) Find the median-median line equation and graph it. (d) Writing to Learn For these data, which of the two lines appears to be the line of better fit? Why?

Suppose \(b^{2}-4 a c>0\) for the equation \(a x^{2}+b x+c=0\). (a) Prove that the sum of the two solutions of this equation is -b/a. (b) Prove that the product of the two solutions of this equation is c/a.

Connecting Algebra and Geometry Prove that the axis of the graph of ƒ(x) = (x - a)(x - b) is x = (a + b)/2, where a and i are real numbers.

Connecting Algebra and Geometry Identify the vertex of the graph of ƒ(x) = (x - a)(x - b), where a and b are any real numbers.

Connecting Algebra and Geometry Prove that if \(x_{1} \text { and } x_{2}\) are real numbers and are zeros of the quadratic function \(f(x)=a x^{2}+b x+c\), then the axis of the graph of ƒ is \(x=\left(x_{1}+x_{2}\right) / 2\).

Prove the Constant Rate of Change Theorem, which is stated on page 160.