Graph the lines on the same coordinate plane, and find the coordinates of the points of

Plot the points , , , , , and on a coordinate plane.

Plot the points , , , , and on a coordinate plane. Draw the line segments AB, BC, CD, DE, and EA.

Plot the points , , , , and . Describe the set of all points of the form , where a is a real number.

Plot the points , , , , and . Describe the set of all points of the form , where a is a real number.

Find the coordinates of the points AF.

Find the coordinates of the points AF.

Describe the set of all points in a coordinate plane that satisfy the given condition. (a) (b) (c) (d) (e) (f)

Describe the set of all points in a coordinate plane that satisfy the given condition. (a) (b) (c) (d) (e) (f)

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

(a) Find the distance between A and B. (b) Find the midpoint of the segment AB.

Show that the triangle with vertices A, B, and C is a right triangle, and find its area.

Show that the triangle with vertices A, B, and C is a right triangle, and find its area.

Show that , , , and are vertices of a square.

Show that , , , and are vertices of a parallelogram.

Given , find the coordinates of the point B such that is the midpoint of segment AB

Given and , find the point on segment AB that is three-fourths of the way from A to B.

Prove that C is on the perpendicular bisector of segment AB.

Prove that C is on the perpendicular bisector of segment AB.

Find a formula that expresses the fact that an arbitrary point is on the perpendicular bisector l of segment AB.

Find a formula that expresses the fact that an arbitrary point is on the perpendicular bisector l of segment AB.

Find a formula that expresses the fact that is a distance 5 from the origin. Describe the set of all such points.

Find a formula that states that is a distance from a fixed point . Describe the set of all such points.

Find all points on the y-axis that are a distance 6 from

Find all points on the x-axis that are a distance 5 from

Find the point with coordinates of the form that is in the third quadrant and is a distance 5 from

Find all points with coordinates of the form that are a distance 3 from

For what values of a is the distance between and greater than

Given and , find a formula not containing radicals that expresses the fact that the sum of the distances from to

Prove that the midpoint of the hypotenuse of any right triangle is equidistant from the vertices. (Hint: Label the vertices of the triangle , , and

Prove that the diagonals of any parallelogram bisect each other. (Hint: Label three of the vertices of the parallelogram , , and .)

Plot the points in the given viewing rectangle.

Plot the points in the given viewing rectangle.

Households with a computer The table lists the number of U.S. households with a computer for selected years. (a) Plot the data in the viewing rectangle by . (b) Discuss how the number of households is changing.

Published newspapers The table lists the number of daily newspapers published in the United States for various years. (a) Plot the data in the viewing rectangle by . (b) Use the midpoint formula to estimate the number of newspapers in 1930. Compare your answer with the true value, which is 1942.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Use tests for symmetry to determine which graphs in the indicated exercises are symmetric with respect to (a) the y-axis, (b) the x-axis, and (c) the origin.The odd-numbered exercises in 120

Use tests for symmetry to determine which graphs in the indicated exercises are symmetric with respect to (a) the y-axis, (b) the x-axis, and (c) the origin.The even-numbered exercises in 120

Complete the statements. (a) As x 1, f(x) ____ (b) As x 2, f(x) ____ (c) As x 3, f(x) ____ (d) As x , f(x) ____ (e) As x , f(x) ____

Complete the statements. (a) As x 2, f(x) ____ (b) As x 1, f(x) ____ (c) As x 0, f(x) ____ (d) As x , f(x) ____ (e) As x , f(x) ____

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Sketch the graph of the circle or semicircle.

Find an equation of the circle that satisfies the stated conditions. Center , radius 5

Find an equation of the circle that satisfies the stated conditions. Center , radius 3

Find an equation of the circle that satisfies the stated conditions. Center , radius

Find an equation of the circle that satisfies the stated conditions. Center , radius

Find an equation of the circle that satisfies the stated conditions. Center , passing through

Find an equation of the circle that satisfies the stated conditions. Center at the origin, passing through

Find an equation of the circle that satisfies the stated conditions. Center , tangent to the y-axis

Find an equation of the circle that satisfies the stated conditions. Center , tangent to the x-axis

Find an equation of the circle that satisfies the stated conditions. Tangent to both axes, center in the second quadrant, radius 2

Find an equation of the circle that satisfies the stated conditions. Tangent to both axes, center in the fourth quadrant, radius 3

Find an equation of the circle that satisfies the stated conditions. Endpoints of a diameter and

Find an equation of the circle that satisfies the stated conditions. Endpoints of a diameter and

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find the center and radius of the circle with the given equation.

Find equations for the upper half, lower half, right half, and left half of the circle.

Find equations for the upper half, lower half, right half, and left half of the circle.

Find equations for the upper half, lower half, right half, and left half of the circle.

Find equations for the upper half, lower half, right half, and left half of the circle.

Find an equation for the circle or semicircle.

Find an equation for the circle or semicircle.

Find an equation for the circle or semicircle.

Find an equation for the circle or semicircle.

Determine whether the point P is inside, outside, or on the circle with center C and radius r.

Determine whether the point P is inside, outside, or on the circle with center C and radius r.

For the given circle, find (a) the x-intercepts and (b) the y-intercepts.

For the given circle, find (a) the x-intercepts and (b) the y-intercepts.

Find an equation of the circle that is concentric (has the same center) with and passes through .

Radio broadcasting ranges The signal from a radio station has a circular range of 50 miles. A second radio station, located 100 miles east and 80 miles north of the first station, has a range of 80 miles. Are there locations where signals can be received from both radio stations? Explain your answer.

A circle of radius 5 has its center at the origin. Inside this circle there is a first-quadrant circle of radius 2 that is tangent to . The y-coordinate of the center of is 2. Find the x-coordinate of the center of .

A circle of radius 5 has its center at the origin. Outside this circle is a first-quadrant circle of radius 2 that is tangent to . The y-coordinate of the center of is 3. Find the x-coordinate of the center of .

Express, in interval form, the x-values such that . Assume all points of intersection are shown on the interval

Express, in interval form, the x-values such that . Assume all points of intersection are shown on the interval

Express, in interval form, the x-values such that . Assume all points of intersection are shown on the interval

Express, in interval form, the x-values such that . Assume all points of intersection are shown on the interval

Graph the unit circle using the equations and in the given viewing rectangle. Then discuss how the viewing rectangle affects the graph, and determine the viewing rectangle that results in a graph that most looks like a circle. (1) by (2) by (3) by (4)

Graph the equation , using the equations and in the viewing rectangle by . (a) Find the number of x- and y-intercepts. (b) Use the graph to determine the region where .

Graph the equation, and estimate the x-intercepts.

Graph the equation, and estimate the x-intercepts.

Graph the two equations on the same coordinate plane, and estimate the coordinates of their points of intersection.

Graph the two equations on the same coordinate plane, and estimate the coordinates of their points of intersection.

Graph the two equations on the same coordinate plane, and estimate the coordinates of their points of intersection.

Graph the two equations on the same coordinate plane, and estimate the coordinates of their points of intersection.

Distance between cars The distance D (in miles) between two cars meeting on the same highway at time t (in minutes) is described by the equation on the interval . Graph D, and describe the motion of the cars.

Water in a pool The amount of water A in a swimming pool on day x is given by , where A is in gallons and corresponds to noon on Sunday. Graph A on the interval , and describe the amount of water in the pool.

Speed of sound The speed of sound v in air varies with temperature. It can be calculated in using the equa tion , where T is temperature (in C). (a) Approximate v when . (b) Determine the temperature to the nearest degree, both algebraically and graphically, when the speed of sound is 1000 .

The area A of an equilateral triangle with a side of length s is . Suppose that A must be equal to with an error of at most . Determine graphically how accurately s must be measured in order to satisfy this error requirement. (Hint: Graph , , and .)

Sketch the line through A and B, and find its slope m.

Sketch the line through A and B, and find its slope m.

Sketch the line through A and B, and find its slope m.

Sketch the line through A and B, and find its slope m.

Sketch the line through A and B, and find its slope m.

Sketch the line through A and B, and find its slope m.

Use slopes to show that the points are vertices of the specified polygon. ; parallelogram

Use slopes to show that the points are vertices of the specified polygon. ; trapezoid

Use slopes to show that the points are vertices of the specified polygon. ; rectangle

Use slopes to show that the points are vertices of the specified polygon. ; right triangle

If three consecutive vertices of a parallelogram are , , and , find the fourth vertex.

Let , , , and denote the vertices of an arbitrary quadrilateral. Show that the line segments joining midpoints of adjacent sides form a parallelogram.

Sketch the graph of for the given values of m.

Sketch the graph of for the given values of m.

Sketch the graph of the line through P for each value of m.

Sketch the graph of the line through P for each value of m.

Write equations of the lines.

Write equations of the lines.

Sketch the graphs of the lines on the same coordinate plane.

Sketch the graphs of the lines on the same coordinate plane.

Find a general form of an equation of the line through the point A that satisfies the given condition. (a) parallel to the y-axis (b) perpendicular to the y-axis

Find a general form of an equation of the line through the point A that satisfies the given condition. (a) parallel to the x-axis (b) perpendicular to the x-axis

Find a general form of an equation of the line through the point A that satisfies the given condition. slope

Find a general form of an equation of the line through the point A that satisfies the given condition. slope

Find a general form of an equation of the line through the point A that satisfies the given condition. slope

Find a general form of an equation of the line through the point A that satisfies the given condition. slope

Find a general form of an equation of the line through the point A that satisfies the given condition. through

Find a general form of an equation of the line through the point A that satisfies the given condition. x-intercept

Find a general form of an equation of the line through the point A that satisfies the given condition. parallel to the line

Find a general form of an equation of the line through the point A that satisfies the given condition. parallel to the line

Find a general form of an equation of the line through the point A that satisfies the given condition. perpendicular to the line

Find a general form of an equation of the line through the point A that satisfies the given condition. perpendicular to the line

Find the slope-intercept form of the line that satisfies the given conditions. x-intercept y-intercept

Find the slope-intercept form of the line that satisfies the given conditions. x-intercept , y-intercept

Find the slope-intercept form of the line that satisfies the given conditions. Through and

Find the slope-intercept form of the line that satisfies the given conditions. Find the slope-intercept form of the line that satisfies the given conditions.

Find a general form of an equation for the perpendicular bisector of the segment AB. ,

Find a general form of an equation for the perpendicular bisector of the segment AB.

Find an equation for the line that bisects the given quadrants. II and IV

Find an equation for the line that bisects the given quadrants. I and III

Use the slope-intercept form to find the slope and y-intercept of the given line, and sketch its graph.

Use the slope-intercept form to find the slope and y-intercept of the given line, and sketch its graph.

Use the slope-intercept form to find the slope and y-intercept of the given line, and sketch its graph.

Use the slope-intercept form to find the slope and y-intercept of the given line, and sketch its graph.

Find an equation of the line shown in the figure.

Find an equation of the line shown in the figure.

If a line l has nonzero x- and y-intercepts a and b, respectively, then its intercept form is Find the intercept form for the given line.

If a line l has nonzero x- and y-intercepts a and b, respectively, then its intercept form is Find the intercept form for the given line.

Find an equation of the circle that has center and is tangent to the line

Find an equation of the line that is tangent to the circle at the point .

Fetal growth The growth of a fetus more than 12 weeks old can be approximated by the formula , where L is the length (in centimeters) and t is the age (in weeks). Prenatal length can be determined by ultrasound. Approximate the age of a fetus whose length is 28 centimeters.

Estimating salinity Salinity of the ocean refers to the amount of dissolved material found in a sample of seawater. Salinity S can be estimated from the amount C of chlorine in seawater using , where S and C are measured by weight in parts per thousand. Approximate C if S is 0.35.

Weight of a humpback whale The expected weight W (in tons) of a humpback whale can be approximated from its length L (in feet) by using 42.8 for . (a) Estimate the weight of a 40-foot humpback whale. (b) If the error in estimating the length could be as large as 2 feet, what is the corresponding error for the weight estimate?

Growth of a blue whale Newborn blue whales are approximately 24 feet long and weigh 3 tons. Young whales are nursed for 7 months, and by the time of weaning they often are 53 feet long and weigh 23 tons. Let L and W denote the length (in feet) and the weight (in tons), respectively, of a whale that is t months of age. (a) If L and t are linearly related, express L in terms of t. (b) What is the daily increase in the length of a young whale? (Use 1 days.) (c) If W and t are linearly related, express W in terms of t. (d) What is the daily increase in the weight of a young whale?

Baseball stats Suppose a major league baseball player has hit 5 home runs in the first 14 games, and he keeps up this pace throughout the 162-game season.(a) Express the number y of home runs in terms of the number x of games played. (b) How many home runs will the player hit for the season?

Cheese production A cheese manufacturer produces 18,000 pounds of cheese from January 1 through March 24. Suppose that this rate of production continues for the remainder of the year. (a) Express the number y of pounds of cheese produced in terms of the number x of the day in a 365-day year. (b) Predict, to the nearest pound, the number of pounds produced for the year.

Childhood weight A baby weighs 10 pounds at birth, and three years later the childs weight is 30 pounds. Assume that childhood weight W (in pounds) is linearly related to age t (in years). (a) Express W in terms of t. (b) What is W on the childs sixth birthday? (c) At what age will the child weigh 70 pounds? (d) Sketch, on a tW-plane, a graph that shows the relationship between W and t for .

Loan repayment A college student receives an interestfree loan of $8250 from a relative. The student will repay $125 per month until the loan is paid off. (a) Express the amount P (in dollars) remaining to be paid in terms of time t (in months). (b) After how many months will the student owe $5000? (c) Sketch, on a tP-plane, a graph that shows the relationship between P and t for the duration of the loan.

Vaporizing water The amount of heat H (in joules) required to convert one gram of water into vapor is linearly related to the temperature T (in C) of the atmosphere. At 10C this conversion requires 2480 joules, and each increase in temperature of 15C lowers the amount of heat needed by 40 joules. Express H in terms of T.

Aerobic power In exercise physiology, aerobic power P is defined in terms of maximum oxygen intake. For altitudes up to 1800 meters, aerobic power is optimalthat is, 100%. Beyond 1800 meters, P decreases linearly from the maximum of 100% to a value near 40% at 5000 meters. (a) Express aerobic power P in terms of altitude h (in meters) for . (b) Estimate aerobic power in Mexico City (altitude: 2400 meters), the site of the 1968 Summer Olympic Games.

Urban heat island The urban heat island phenomenon has been observed in Tokyo. The average temperature was 13.5C in 1915, and since then has risen 0.032C per year. (a) Assuming that temperature T (in C) is linearly related to time t (in years) and that corresponds to 1915, express T in terms of t. (b) Predict the average temperature in the year 2020.

Rising ground temperature In 1870 the average ground temperature in Paris was 11.8C. Since then it has risen at a nearly constant rate, reaching 13.5C in 1969. (a) Express the temperature T (in C) in terms of time t (in years), where corresponds to the year 1870 and . (b) During what year was the average ground temperature 12.5C?

Business expenses The owner of an ice cream franchise must pay the parent company $1000 per month plus 5% of the monthly revenue R. Operating cost of the franchise includes a fixed cost of $2600 per month for items such as utilities and labor. The cost of ice cream and supplies is 50% of the revenue. (a) Express the owners monthly expense E in terms of R. (b) Express the monthly profit P in terms of R. (c) Determine the monthly revenue needed to break even.

Drug dosage Pharmacological products must specify recommended dosages for adults and children. Two formulas for modification of adult dosage levels for young children are Cowlings rule: and Friends rule: where a denotes adult dose (in milligrams) and t denotes the age of the child (in years). (a) If , graph the two linear equations on the same coordinate plane for . (b) For what age do the two formulas specify the same dosage?

Video game In the video game shown in the figure, an airplane flies from left to right along the path given by and shoots bullets in the tangent direction at creatures placed along the x-axis at , 2, 3, 4. From calculus, the slope of the tangent line to the path at is and at is Determine whether a creature will be hit if bullets are shot when the airplane is at (a) P (b) Q

Temperature scales The relationship between the temperature reading F on the Fahrenheit scale and the temperature reading C on the Celsius scale is given by . (a) Find the temperature at which the reading is the same on both scales. (b) When is the Fahrenheit reading twice the Celsius reading?

Vertical wind shear Vertical wind shear occurs when wind speed varies at different heights above the ground. Wind shear is of great importance to pilots during takeoffs and landings. If the wind speed is at height and at height , then the average wind shear s is given by the slope formula If the wind speed at ground level is 22 and s has been determined to be 0.07, find the wind speed 185 feet above the ground.

Vertical wind shear In the study of vertical wind shear, the formula is sometimes used, where P is a variable that depends on the terrain and structures near ground level. In Montreal, the average daytime value for P with north winds over 29 was determined to be 0.13. If a 32 north wind is measured 20 feet above the ground, approximate the average wind shear (see Exercise 67) between 20 feet and 200 feet.

The given points were found using empirical methods. Determine whether they lie on the same line , and if so, find the values of a and b.

The given points were found using empirical methods. Determine whether they lie on the same line , and if so, find the values of a and b.

Graph the lines on the same coordinate plane, and find the coordinates of the points of intersection (the coordinates are integers).

Graph the lines on the same coordinate plane, and find the coordinates of the points of intersection (the coordinates are integers).

Graph the lines on the same coordinate plane, and estimate the coordinates of the points of intersection. Identify the polygon determined by the lines.

Graph the lines on the same coordinate plane, and estimate the coordinates of the points of intersection. Identify the polygon determined by the lines.

For the data table, determine a line in the form that approximately models the data. Plot the line together with the data on the same coordinate axes. Note: For exercises requiring an approximate model, answers may vary depending on the data points selected.

For the data table, determine a line in the form that approximately models the data. Plot the line together with the data on the same coordinate axes. Note: For exercises requiring an approximate model, answers may vary depending on the data points selected.

Record distances in the triple jump World record distances (in meters) for the triple jump are listed in the table.(a) Plot the data. (b) Find a line of the form D  aY  b that approximates these data, where D is the distance and Y is the year. Graph this line together with the data on the same coordinate axes. (c) Use the line to predict the record distance in 1985, and compare it with the actual record of 17.97 meters. (d) Interpret the slope of this line.

Record times in the mile World record times (in seconds) for the mile run are listed in the table. (a) Plot the data. (b) Find a line of the form that approximates these data, where T is the time and Y is the year. Graph this line together with the data on the same coordinate axes. (c) Use the line to predict the record time in 1985, and compare it with the actual record of 226.3 seconds. (d) Interpret the slope of this line.

If , find

If , find

If , find , , a

If , find , , and . f3

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a and h are real numbers, find (a) (b) (c) (d) (e) (f) if

If a is a positive real number, find (a) (b) (c) (d)

If a is a positive real number, find (a) (b) (c) (d)

If a is a positive real number, find (a) (b) (c) (d)

If a is a positive real number, find (a) (b) (c) (d)

Explain why the graph is or is not the graph of a function.

Explain why the graph is or is not the graph of a function.

Determine the domain D and range R of the function shown in the figure.

Determine the domain D and range R of the function shown in the figure.

For the graph of the function f sketched in the figure, determine (a) the domain (b) the range (c) (d) all x such that (e) all x such that

For the graph of the function f sketched in the figure, determine (a) the domain (b) the range (c) (d) all x such that (e) all x such that

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

Find the domain of f.

(a) Find the domain D and range R of f. (b) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Find the domain D and range R of f. (b) Find the intervals on which f is increasing, is decreasing, or is constant.

Sketch the graph of a function that is increasing on and and is decreasing on .

Sketch the graph of a function that is decreasing on and and is increasing on and .

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

Simplify the difference quotient if .

Simplify the difference quotient if .

Simplify the difference quotient if .

Simplify the difference quotient if .

Simplify the difference quotient if .

Simplify the difference quotient if .

If a linear function f satisfies the given conditions, find

If a linear function f satisfies the given conditions, find

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Determine whether the set W of ordered pairs is a function in the sense of the alternative definition of function on page 130.

Constructing a box From a rectangular piece of cardboard having dimensions 20 inches 30 inches, an open box is to be made by cutting out an identical square of area from each corner and turning up the sides (see the figure). Express the volume V of the box as a function of x.

Constructing a storage tank Refer to Example 8. A steel storage tank for propane gas is to be constructed in the shape of a right circular cylinder of altitude 10 feet with a hemisphere attached to each end. The radius r is yet to be determined. Express the surface area S of the tank as a function of r.

Dimensions of a building A small office unit is to contain 500 of floor space. A simplified model is shown in the figure on the next page. (a) Express the length y of the building as a function of the width x. (b) If the walls cost $100 per running foot, express the cost C of the walls as a function of the width x. (Disregard the wall space above the doors and the thickness of the walls.

Dimensions of an aquarium An aquarium of height 1.5 feet is to have a volume of 6 . Let x denote the length of the base and y the width (see the figure). (a) Express y as a function of x. (b) Express the total number S of square feet of glass needed as a function of x.

Skyline ordinance A city council is proposing a new skyline ordinance. It would require the setback S for any building from a residence to be a minimum of 100 feet, plus an additional 6 feet for each foot of height above 25 feet. Find a linear function for S in terms of h.

Energy tax A proposed energy tax T on gasoline, which would affect the cost of driving a vehicle, is to be computed by multiplying the number x of gallons of gasoline that you buy by 125,000 (the number of BTUs per gallon of gasoline) and then multiplying the total BTUs by the tax 34.2 cents per million BTUs. Find a linear function for T in terms of x.

Childhood growth For children between ages 6 and 10, height y (in inches) is frequently a linear function of age t (in years). The height of a certain child is 48 inches at age 6 and 50.5 inches at age 7. (a) Express y as a function of t. (b) Sketch the line in part (a), and interpret the slope. (c) Predict the height of the child at age 10.

Radioactive contamination It has been estimated that 1000 curies of a radioactive substance introduced at a point on the surface of the open sea would spread over an area of 40,000 in 40 days. Assuming that the area covered by the radioactive substance is a linear function of time t and is always circular in shape, express the radius r of the contamination as a function of t.

Distance to a hot-air balloon A hot-air balloon is released at 1:00 P.M. and rises vertically at a rate of 2 . An observation point is situated 100 meters from a point on the ground directly below the balloon (see the figure). If t denotes the time (in seconds) after 1:00 P.M., express the distance d between the balloon and the observation point as a function of t.

Triangle ABC is inscribed in a semicircle of diameter 15 (see the figure). (a) If x denotes the length of side AC, express the length y of side BC as a function of x. (Hint: Angle ACB is a right angle.) (b) Express the area of triangle ABC as a function of x, and state the domain of this function.

Distance to Earth From an exterior point P that is h units from a circle of radius r, a tangent line is drawn to the circle (see the figure). Let y denote the distance from the point P to the point of tangency T. (a) Express y as a function of h. (Hint: If C is the center of the circle, then PT is perpendicular to CT.) (b) If r is the radius of Earth and h is the altitude of a space shuttle, then y is the maximum distance to Earth that an astronaut can see from the shuttle. In particular, if mi and mi, approximate

Length of a tightrope The figure illustrates the apparatus for a tightrope walker. Two poles are set 50 feet apart, but the point of attachment P for the rope is yet to be determined. (a) Express the length L of the rope as a function of the distance x from P to the ground. (b) If the total walk is to be 75 feet, determine the distance from P to the ground.

Airport runway The relative positions of an aircraft runway and a 20-foot-tall control tower are shown in the figure. The beginning of the runway is at a perpendicular distance of 300 feet from the base of the tower. If x denotes the distance an airplane has moved down the runway, express the distance d between the airplane and the top of the control tower as a function of x.

Destination time A man in a rowboat that is 2 miles from the nearest point A on a straight shoreline wishes to reach a house located at a point B that is 6 miles farther down the shoreline (see the figure). He plans to row to a point P that is between A and B and is x miles from the house, and then he will walk the remainder of the distance. Suppose he can row at a rate of 3 and can walk at a rate of 5 . If T is the total time required to reach the house, express T as a function of x.

(a) Sketch the graph of f on the given interval [a, b]. (b) Estimate the range of f on [a, b]. (c) Estimate the intervals on which f is increasing or is decreasing.

(a) Sketch the graph of f on the given interval [a, b]. (b) Estimate the range of f on [a, b]. (c) Estimate the intervals on which f is increasing or is decreasing.

(a) Sketch the graph of f on the given interval [a, b]. (b) Estimate the range of f on [a, b]. (c) Estimate the intervals on which f is increasing or is decreasing.

(a) Sketch the graph of f on the given interval [a, b]. (b) Estimate the range of f on [a, b]. (c) Estimate the intervals on which f is increasing or is decreasing.

In Exercises 4344 of Section 1.4, algebraic methods were used to find solutions to each of the following equations. Now solve the equation graphically by assigning the expression on the left side to and the number on the right side to and then finding the x-coordinates of all points of intersection of the two graphs.

In Exercises 4344 of Section 1.4, algebraic methods were used to find solutions to each of the following equations. Now solve the equation graphically by assigning the expression on the left side to and the number on the right side to and then finding the x-coordinates of all points of intersection of the two graphs.

Calculator screen A particular graphing calculator screen is 95 pixels wide and 63 pixels high. (a) Find the total number of pixels in the screen. (b) If a function is graphed in dot mode, determine the maximum number of pixels that would typically be darkened on the calculator screen to show the function.

Stopping distances The table lists the practical stopping distances D (in feet) for cars at speeds S (in miles per hour) on level surfaces, as used by the American Association of State Highway and Transportation Officials. (a) Plot the data. (b) Determine whether stopping distance is a linear function of speed. (c) Discuss the practical implications of these data for safely driving a car.

New car prices In 1993 and 2000, the average prices paid for a new car were $16,871 and $20,356, respectively. Assume the average price increased linearly. (a) Find a function f that models the average price paid for a new car. Graph f together with the two data points. (b) Interpret the slope of the graph of f. (c) Graphically approximate the year when the average price paid would be $25,000.

Suppose f is an even function and g is an odd function. Complete the table, if possible.

Suppose f is an even function and g is an odd function. Complete the table, if possible.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Determine whether f is even, odd, or neither even nor odd.

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

Sketch, on the same coordinate plane, the graphs of f for the given values of c. (Make use of symmetry, shifting, stretching, compressing, or reflecting.)

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

If the point P is on the graph of a function f, find the corresponding point on the graph of the given function.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

Explain how the graph of the function compares to the graph of . For example, for the equation , the graph of f is shifted 3 units to the left and stretched vertically by a factor of 2.

The graph of a function f with domain is shown in the figure. Sketch the graph of the given equation.

The graph of a function f with domain is shown in the figure. Sketch the graph of the given equation.

The graph of a function f is shown, together with graphs of three other functions (a), (b), and (c). Use properties of symmetry, shifts, and reflecting to find equations for graphs (a), (b), and (c) in terms of f.

The graph of a function f is shown, together with graphs of three other functions (a), (b), and (c). Use properties of symmetry, shifts, and reflecting to find equations for graphs (a), (b), and (c) in terms of f.

The graph of a function f is shown, together with graphs of three other functions (a), (b), and (c). Use properties of symmetry, shifts, and reflecting to find equations for graphs (a), (b), and (c) in terms of f.

The graph of a function f is shown, together with graphs of three other functions (a), (b), and (c). Use properties of symmetry, shifts, and reflecting to find equations for graphs (a), (b), and (c) in terms of f.

Sketch the graph of

Sketch the graph of

Sketch the graph of

Sketch the graph of

Sketch the graph of

Sketch the graph of

The symbol denotes values of the greatest integer function. Sketch the graph of f. a) (b) (c) (d) (e)

The symbol denotes values of the greatest integer function. Sketch the graph of f.a) (b) (c) (d) (e)

Complete the statements.

Complete the statements.

Explain why the graph of the equation is not the graph of a function.

Explain why the graph of the equation is not the graph of a function.

For the graph of shown in the figure, sketch the graph of .

For the graph of shown in the figure, sketch the graph of .

Sketch the graph of the equation.

Sketch the graph of the equation.

Sketch the graph of the equation.

Sketch the graph of the equation.

Let be a function with domain and range . Find the domain D and range R for each function. Assume and . (a) (b) (c) (d) (e) (f) (g) (h)

Let be a function with domain and range . Find the domain D and range R for each function. (a) (b) (c) (d) (e) (f) (g) (h)

Income tax rates A certain country taxes the first $20,000 of an individuals income at a rate of 15%, and all income over $20,000 is taxed at 20%. Find a piecewise-defined function T that specifies the total tax on an income of x dollars.

Property tax rates A certain state taxes the first $600,000 in property value at a rate of 1%; all value over $600,000 is taxed at 1.25%. Find a piecewise-defined function T that specifies the total tax on a property valued at x dollars.

Royalty rates A certain paperback sells for $12. The author is paid royalties of 10% on the first 10,000 copies sold, 12.5% on the next 5000 copies, and 15% on any additional copies. Find a piecewise-defined function R that specifies the total royalties if x copies are sold.

Electricity rates An electric company charges its customers $0.0577 per kilowatt-hour (kWh) for the first 1000 kWh used, $0.0532 for the next 4000 kWh, and $0.0511 for any kWh over 5000. Find a piecewise-defined function C for a customers bill of x kWh.

Estimate the solutions of the inequality.

Estimate the solutions of the inequality.

Estimate the solutions of the inequality.

Estimate the solutions of the inequality.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Graph f in the viewing rectangle by . Use the graph of f to predict the graph of g. Verify your prediction by graphing g in the same viewing rectangle.

Car rental charges There are two car rental options available for a four-day trip. Option I is $45 per day, with 200 free miles and $0.40 per mile for each additional mile. Option II is $58.75 per day, with a charge of $0.25 per mile. (a) Determine the cost of a 500-mile trip for both options. (b) Model the data with a cost function for each fourday option. (c) Make a table that lists the mileage and the charge for each option for trips between 100 and 1200 miles, using increments of 100 miles. (d) Use the table to determine the mileages at which each option is preferable.

Traffic flow Cars are crossing a bridge that is 1 mile long. Each car is 12 feet long and is required to stay a distance of at least d feet from the car in front of it (see figure). (a) Show that the largest number of cars that can be on the bridge at one time is , where denotes the greatest integer function. (b) If the velocity of each car is v , show that the maximum traffic flow rate F (in ) is given by .

Find the standard equation of any parabola that has vertex V.

Find the standard equation of any parabola that has vertex V.

Find the standard equation of any parabola that has vertex V.

Find the standard equation of any parabola that has vertex V.

Express in the form

Express in the form

Express in the form

Express in the form

Express in the form

Express in the form

Express in the form

Express in the form

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

(a) Use the quadratic formula to find the zeros of f. (b) Find the maximum or minimum value of . (c) Sketch the graph of f.

Find the standard equation of the parabola shown in the figure.

Find the standard equation of the parabola shown in the figure.

Find the standard equation of the parabola shown in the figure.

Find the standard equation of the parabola shown in the figure.

Find an equation of the form of the parabola shown in the figure.

Find an equation of the form of the parabola shown in the figure.

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex , passing through

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex , passing through

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex , x-intercept

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. Vertex , x-intercept

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. x-intercepts and 5, highest point has y-coordinate

Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions. -intercepts 8 and 0, lowest point has y-coordinate

Find the maximum vertical distance d between the parabola and the line for the green region.

Find the maximum vertical distance d between the parabola and the line for the green region.

Ozone occurs at all levels of Earths atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density of ozone (in ) for altitudes h between 20 kilometers and 35 kilometers was determined experimentally. For each and season, approximate the altitude at which the density of ozone is greatest. (autumn)

Ozone occurs at all levels of Earths atmosphere. The density of ozone varies both seasonally and latitudinally. At Edmonton, Canada, the density of ozone (in ) for altitudes h between 20 kilometers and 35 kilometers was determined experimentally. For each and season, approximate the altitude at which the density of ozone is greatest. (spring)

Infant growth rate The growth rate y (in pounds per month) of an infant is related to present weight x (in pounds) by the formula , where c is a positive constant and . At what weight does the maximum growth rate occur?

Gasoline mileage The number of miles M that a certain automobile can travel on one gallon of gasoline at a speed of v is given by for (a) Find the most economical speed for a trip. (b) Find the largest value of M.

Height of a projectile An object is projected vertically upward from the top of a building with an initial velocity of 144 . Its distance in feet above the ground after t seconds is given by the equation (a) Find its maximum distance above the ground. (b) Find the height of the building.

Flight of a projectile An object is projected vertically upward with an initial velocity of , and its distance in feet above the ground after t seconds is given by the formula . (a) If the object hits the ground after 12 seconds, find its initial velocity . (b) Find its maximum distance above the ground.

Find two positive real numbers whose sum is 40 and whose product is a maximum.

Find two real numbers whose difference is 60 and whose product is a minimum.

Constructing cages One thousand feet of chain-link fence is to be used to construct six animal cages, as shown in the figure. (a) Express the width y as a function of the length x. (b) Express the total enclosed area A of the cages as a function of x. (c) Find the dimensions that maximize the enclosed area.

Fencing a field A farmer wishes to put a fence around a rectangular field and then divide the field into three rectangular plots by placing two fences parallel to one of the sides. If the farmer can afford only 1000 yards of fencing, what dimensions will give the maximum rectangular area?

Leaping animals Flights of leaping animals typically have parabolic paths. The figure illustrates a frog jump superimposed on a coordinate plane. The length of the leap is 9 feet, and the maximum height off the ground is 3 feet. Find a standard equation for the path of the frog.

The human cannonball In the 1940s, the human cannonball stunt was performed regularly by Emmanuel Zacchini for The Ringling Brothers and Barnum & Bailey Circus. The tip of the cannon rose 15 feet off the ground, and the total horizontal distance traveled was 175 feet. When the cannon is aimed at an angle of 45, an equation of the parabolic flight (see the figure) has the form (a) Use the given information to find an equation of the flight. (b) Find the maximum height attained by the human cannonball.

Shape of a suspension bridge One section of a suspension bridge has its weight uniformly distributed between twin towers that are 400 feet apart and rise 90 feet above the horizontal roadway (see the figure). A cable strung between the tops of the towers has the shape of a parabola, and its center point is 10 feet above the roadway. Suppose coordinate axes are introduced, as shown in the figure. (a) Find an equation for the parabola. (b) Nine equally spaced vertical cables are used to support the bridge (see the figure). Find the total length of these supports.

Designing a highway Traffic engineers are designing a stretch of highway that will connect a horizontal highway with one having a 20% grade that is, slope , as illustrated in the figure. The smooth transition is to take place over a horizontal distance of 800 feet, with a parabolic piece of highway used to connect points A and B. If the equation of the parabolic segment is of the form , it can be shown that the slope of the tangent line at the point on the parabola is given by . (a) Find an equation of the parabola that has a tangent line of slope 0 at A and at B. (b) Find the coordinates of B.

Parabolic doorway A doorway has the shape of a parabolic arch and is 9 feet high at the center and 6 feet wide at the base. If a rectangular box 8 feet high must fit through the doorway, what is the maximum width the box can have?

Path of a baseball Assume a baseball hit at home plate follows a parabolic path having equation where x and y are both measured in feet. (a) Find the maximum height of the baseball. (b) Does the baseball clear an 8-foot fence that is 385 feet from home plate?

Quantity discount A company sells running shoes to dealers at a rate of $40 per pair if fewer than 50 pairs are ordered. If a dealer orders 50 or more pairs (up to 600), the price per pair is reduced at a rate of 4 cents times the number ordered. What size order will produce the maximum amount of money for the company?

Group discount A travel agency offers group tours at a rate of $60 per person for the first 30 participants. For larger groupsup to 90each person receives a $0.50 discount for every participant in excess of 30. For example, if 31 people participate, then the cost per person is $59.50. Determine the size of the group that will produce the maximum amount of money for the agency.

Cable TV fee A cable television firm presently serves 8000 households and charges $50 per month. A marketing survey indicates that each decrease of $5 in the monthly charge will result in 1000 new customers. Let denote the total monthly revenue when the monthly charge is x dollars. (a) Determine the revenue function R. (b) Sketch the graph of R and find the value of x that results in maximum monthly revenue.

Apartment rentals A real estate company owns 218 efficiency apartments, which are fully occupied when the rent is $940 per month. The company estimates that for each $25 increase in rent, 5 apartments will become unoccupied. What rent should be charged so that the company will receive the maximum monthly income?

Graph and f on the same coordinate plane, and estimate the points of intersection.

Graph and f on the same coordinate plane, and estimate the points of intersection.

Graph, on the same coordinate plane, for , and 4, and describe how the value of a affects the graph.

Graph, on the same coordinate plane, for , and , and describe how the value of b affects the graph.

Precipitation in Seattle The average monthly precipitation (in inches) for Seattle is listed in the following table. (Note: April average is not given.) (a) Plot the average monthly precipitation. (b) Model the data with a quadratic function of the form . Graph f and the data on the same coordinate axes. (c) Use f to predict the average rainfall in April. Compare your prediction with the actual value of 2.55 inches.

Handgun homicides The annual numbers of handgun homicides (in thousands) from 1982 to 1993 are listed in the table. (After this period, the number of handgun homicides decreased and leveled off to values similar to those in the mid-1980s.) (a) Plot the data. Discuss any overall trends in the data. (b) Model these data with a quadratic function of the form . (c) Graph f together with the data.

Crest vertical curves When engineers plan highways, they must design hills so as to ensure proper vision for drivers. Hills are referred to as crest vertical curves. Crest vertical curves change the slope of a highway. Engineers use a parabolic shape for a highway hill, with the vertex located at the top of the crest. Two roadways with different slopes are to be connected with a parabolic crest curve. The highway passes through the points , , , , and , as shown in the figure. The roadway is linear between A and B, parabolic between B and D, and then linear between D and E. (a) Find a piecewise-defined function f that models the roadway between the points A and E. (b) Graph f in the viewing rectangle by .

Sag vertical curves Refer to Exercise 63. Valleys or dips in highways are referred to as sag vertical curves. Sag vertical curves are also modeled using parabolas. Two roadways with different grades meeting at a sag curve need to be connected. The highway passes through the points , , , , and , as shown in the figure. The roadway is linear between A and B, parabolic between B and D, and linear between D and E. (a) Find a piecewise-defined function f that models the roadway between the points A and E. (b) Graph f in the viewing rectangle by .

Parabolic path Under ideal conditions an object thrown from level ground will follow a parabolic path of the form , where a and b are constants and x represents the horizontal distance traveled by the object. (a) Determine a and b so that the object reaches a maximum height of 100 feet and travels a horizontal distance of 150 feet before striking the ground. (b) Graph in the viewing rectangle by . (c) Graph , where , in the same viewing rectangle of by . How does the constant k affect the path of the object?

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) , , , and (b) the domain of , , and fg (c) the domain of

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) (b) (c) (d)

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Find (a) and the domain of and (b) and the domain of

Solve the equation .

Solve the equation .

Several values of two functions f and g are listed in the following tables: If possible, find (a) (b) (c) (d) (e)

Several values of two functions T and S are listed in the following tables: If possible, find (a) (b) (c) (d) (e)

If and , find

If and , find

If f is an odd function and g is an even function, is fg even, odd, or neither even nor odd?

There is one function with domain that is both even and odd. Find that function.

Payroll functions Let the social security tax function SSTAX be defined as , where is the weekly income. Let ROUND2 be the function that rounds a number to two decimal places. Find the value of

Computer science functions Let the function CHR be defined by , . Then let the function ORD be defined by . Find

Spreading fire A fire has started in a dry open field and is spreading in the form of a circle. If the radius of this circle increases at the rate of 5 , express the total fire area A as a function of time t (in minutes).

Dimensions of a balloon A spherical balloon is being inflated at a rate of . Express its radius r as a function of time t (in minutes), assuming that when .

Dimensions of a sand pile The volume of a conical pile of sand is increasing at a rate of , and the height of the pile always equals the radius r of the base. Express r as a function of time t (in minutes), assuming that when .

Diagonal of a cube The diagonal d of a cube is the distance between two opposite vertices. Express d as a function of the edge x of the cube. (Hint: First express the diagonal y of a face as a function of x.)

Altitude of a balloon A hot-air balloon rises vertically from ground level as a rope attached to the base of the balloon is released at the rate of 5 (see the figure). The pulley that releases the rope is 20 feet from a platform where passengers board the balloon. Express the altitude h of the balloon as a function of time t.

Tightrope walker Refer to Exercise 76 of Section 2.4. Starting at the lowest point, the tightrope walker moves up the rope at a steady rate of 2 . If the rope is attached 30 feet up the pole, express the height h of the walker above the ground as a function of time t. (Hint: Let d denote the total distance traveled along the wire. First express d as a function of t, and then h as a function of d.)

Airplane take-off Refer to Exercise 77 of Section 2.4. When the airplane is 500 feet down the runway, it has reached a speed of 150 (or about 102 ), which it will maintain until take-off. Express the distance d of the plane from the control tower as a function of time t (in seconds). (Hint: In the figure, first write x as a function of t.)

Cable corrosion A 100-foot-long cable of diameter 4 inches is submerged in seawater. Because of corrosion, the surface area of the cable decreases at the rate of 750 per year. Express the diameter d of the cable as a function of time t (in years). (Disregard corrosion at the ends of the cable.)

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

Find a composite function form for y.

If and , approximate . In order to avoid calculating a zero value for , rewrite the formula for as

If and , approximate

Refer to Exercise 65 of Section 2.5. Make the assignments and . Determine assignments for (and if necessary) that will enable you to graph each function in (a)(h), and then graph the function. (Check the domain and range with the previously listed answer.) (a) (b) (c) (d) (e) (f) (g) (h)

Refer to Exercise 66 of Section 2.5. Make the assignments and . Determine assignments for and that will enable you to graph each function, and then graph the function. (a) (b) (c) (d) (e) (f) (g) (h)

Describe the set of all points in a coordinate plane such that

Show that the triangle with vertices , , and is a right triangle, and find its area.

Given and , find (a) the distance (b) the midpoint of the segment PQ (c) a point R such that Q is the midpoint of PR

Find all points on the y-axis that are a distance 13 from .

For what values of a is the distance between and less than 3?

Find an equation of the circle that has center and passes through

Find an equation of the circle that has endpoints of a diameter and

Find an equation for the left half of the circle given by .

Find the slope of the line through and

Show that , , , and are vertices of a trapezoid.

Find an equation of the line through that is (a) parallel to the line (b) perpendicular to the line 6x  2y  5 

Express in slope-intercept form

Find an equation of the circle that has center and is tangent to the line

Find an equation of the line that has x-intercept and passes through the center of the circle that has equation

Find a general form of an equation of the line through with slope 4.

Given and , find a general form of an equation for the perpendicular bisector of segment AB.

Find the center and radius of the circle with the given equation. x2  y2  12y  31  0

Find the center and radius of the circle with the given equation. 4x2  4y2  24x  16y  41  0

If , find (a) (b) (c) (d) (e) (f) (g)

Find the sign of without actually finding .

Find the sign of without actually finding .

Find the domain and range of f if (a) (b)

Find if h  0.

Find if h  0.

Find a linear function f such that and

Determine whether f is even, odd, or neither even nor odd. (a) (b) (c)

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Sketch the graph of the equation, and label the x- and y-intercepts.

Find the center of the small circle.

Explain how the graph of compares to the graph of .

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

(a) Sketch the graph of f. (b) Find the domain D and range R of f. (c) Find the intervals on which f is increasing, is decreasing, or is constant.

Sketch the graphs of the following equations, making use of shifting, stretching, or reflecting: (a) (b) (c) (d) (e) (f)

The graph of a function f with domain is shown in the figure. Sketch the graph of the given equation. (a) (b) (c) (d) (e) (f) (g)

Find an equation for the graph shown in the figure.

Find an equation for the graph shown in the figure.

Find an equation for the graph shown in the figure.

Find an equation for the graph shown in the figure.

Find the maximum or minimum value of .

Find the maximum or minimum value of .

Find the maximum or minimum value of .

Find the maximum or minimum value of .

Express the function in the form .

Find the standard equation of a parabola with a vertical axis that has vertex and passes through .

If and , find the domain of (a) fg (b)

If and , find (a) (b)

Find (a) and (b) .

Find (a) and (b) .

Find (a) and the domain of and (b) and the domain of .

Find (a) and the domain of and (b) and the domain of .

Find a composite function form for .

Wheelchair ramp The Americans with Disabilities Act of 1990 guarantees all persons the right of accessibility of public accommodations. Providing access to a building often involves building a wheelchair ramp. Ramps should have approximately 1 inch of vertical rise for every 1220 inches of horizontal run. If the base of an exterior door is located 3 feet above a sidewalk, determine the range of appropriate lengths for a wheelchair ramp.

Discus throw Based on Olympic records, the winning distance for the discus throw can be approximated by the equation , where d is in feet and corresponds to the year 1948. (a) Predict the winning distance for the Summer Olympics in the year 2016. (b) Estimate the Olympic year in which the winning distance will be 265 feet.

House appreciation Six years ago a house was purchased for $179,000. This year it was appraised at $215,000. Assume that the value V of the house after its purchase is a linear function of time t (in years). (a) Express V in terms of t. (b) How many years after the purchase date was the house worth $193,000?

Temperature scales The freezing point of water is 0C, or 32F, and the boiling point is 100C, or 212F. (a) Express the Fahrenheit temperature F as a linear function of the Celsius temperature C. (b) What temperature increase in F corresponds to an increase in temperature of 1C?

Gasoline mileage Suppose the cost of driving an automobile is a linear function of the number x of miles driven and that gasoline costs $3 per gallon. A certain automobile presently gets 20 , and a tune-up that will improve gasoline mileage by 10% costs $120. (a) Express the cost of driving without a tune-up in terms of x. (b) Express the cost of driving with a tune-up in terms of x. (c) How many miles must the automobile be driven after a tune-up to make the cost of the tune-up worthwhile?

Dimensions of a pen A pen consists of five congruent rectangles, as shown in the figure. (a) Express the length y as a function of the length x. (b) If the sides cost $10 per running foot, express the cost C of the pen as a function of the length x.

Distance between cars At noon, car A is 10 feet to the right and 20 feet ahead of car B, as shown in the figure. If car A continues at 88 ft/sec (or 60 mi/hr) while car B continues at 66 ft/sec (or 45 mi/hr), express the distance d between the cars as a function of t, where t denotes the number of seconds after noon.

Constructing a storage shelter An open rectangular storage shelter, consisting of two 4-foot-wide vertical sides and a flat roof, is to be attached to an existing structure, as illustrated in the figure. The flat roof is made of tin and costs $5 per square foot, and the two sides are made of plywood costing $2 per square foot. (a) If $400 is available for construction, express the length y as a function of the height x. (b) Express the volume V inside the shelter as a function of x.

Constructing a cylindrical container A company plans to manufacture a container having the shape of a right circular cylinder, open at the top, and having a capacity of . If the cost of the material for the bottom is and that for the curved sides is , express the total cost C of the material as a function of the radius r of the base of the container.

Filling a pool A cross section of a rectangular pool of dimensions 80 feet by 40 feet is shown in the figure. The pool is being filled with water at a rate of 10 .(a) Express the volume V of the water in the pool as a function of time t. (b) Express V as a function of the depth h at the deep end for and then for . (c) Express h as a function of t for and then for .

Filtering water Suppose 5 of water is poured into a conical filter and subsequently drips into a cup, as shown in the figure. Let x denote the height of the water in the filter, and let y denote the height of the water in the cup. (a) Express the radius r shown in the figure as a function of x. (Hint: Use similar triangles.) (b) Express the height y of the water in the cup as a function of x. (Hint: What is the sum of the two volumes shown in the figure?)

Frustum of a cone The shape of the first spacecraft in the Apollo program was a frustum of a right circular conea solid formed by truncating a cone by a plane parallel to its base. For the frustum shown in the figure, the radii a and b have already been determined. (a) Use similar triangles to express y as a function of h. (b) Derive a formula for the volume of the frustum as a function of h. (c) If ft and ft, for what value of h is the volume of the frustum 600 ?

Water usage rates A certain city charges $3.61 per 1000 gallons of water used up to 5000 gallons and $4.17 per 1000 gallons of water used for more than 5000 gallons. Find a piecewise-defined function B that specifies the total bill for water usage of x gallons.

Long jump record In 1991, Mike Powell of the United States set the world long jump record of 8.95 meters. Assume that the path of his flight was parabolic and that the highest point cleared was 1 meter. Find an equation for his path.

Wire rectangle A piece of wire 24 inches long is bent into the shape of a rectangle having width x and length y. (a) Express y as a function of x. (b) Express the area A of the rectangle as a function of x. (c) Show that the area A is greatest if the rectangle is a square.

Distance between ships At 1:00 P.M. ship A is 30 miles due south of ship B and is sailing north at a rate of 15 . If ship B is sailing west at a rate of 10 , find the time at which the distance d between the ships is minimal (see the figure).

Dimensions of a race track The interior of a half-mile race track consists of a rectangle with semicircles at two opposite ends. Find the dimensions that will maximize the area of the rectangle.

Vertical leaps When a particular basketball player leaps straight up for a dunk, the players distance (in feet) off the floor after t seconds is given by the formula , where g is a gravitational constant. (a) If , find the players hang timethat is, the total number of seconds that the player is in the air. (b) Find the players vertical leapthat is, the maximum distance of the players feet from the floor. (c) On the moon, . Rework parts (a) and (b) for the player on the moon.

Trajectory of a rocket A rocket is fired up a hillside, following a path given by . The hillside has slope , as illustrated in the figure. (a) Where does the rocket land? (b) Find the maximum height of the rocket above the ground.

Compare the graphs of , , , , and on the interval . Write a generalization based on what you find out about graphs of equations of the form , where and p and q are positive integers.

Write an expression for if the graph of g is obtained from the graph of by reflecting f about the (a) x-axis (b) y-axis (c) line (d) line

Consider the graph of , where f is given by . Discuss the general shape of g, including its domain and range. Discuss the advantages and disadvantages of graphing g as a composition of the functions and . (Hint: You may want to use the following expressions for f: , ,

Simplify the difference quotient in Exercises 49 and 50 of Section 2.4 for an arbitrary quadratic function of the form

Refer to Example 5 in Section 2.4. Geometrically, what does the expression represent on the graph of f? What do you think it represents if

The midpoint formula could be considered to be the halfway formula since it gives us the point that is of the distance from the point to the point . Develop an m-nth way formula that gives the point that is of the distance from P to Q (assume m and n are positive integers with

Consider the graphs of equations of the quadratic form that have two x-intercepts. Let d denote the distance from the axis of the parabola to either of the x-intercepts, and let h denote the value of the y-coordinate of the vertex. Explore the relationship between d and h for several specific equations, and then develop a formula for this relationship.

Billing for service A common method of billing for service calls is to charge a flat fee plus an additional fee for each quarter-hour spent on the call. Create a function for a washer repair company that charges $40 plus $20 for each quarter-hour or portion thereoffor example, a 30-minute repair call would cost $80, while a 31-minute repair call would cost $100. The input to your function is any positive integer. (Hint: See Exercise 54(e) of Section 2.5.)

Density of the ozone layer The density D (in ) of the ozone layer at altitudes x between 3 and 15 kilometers during winter at Edmonton, Canada, was determined experimentally to be Express x as a function of D

Precipitation in Minneapolis The average monthly precipitation in inches in Minneapolis is listed in the table. (a) Plot the average monthly precipitation. (b) Model these data with a piecewise function f that is first quadratic and then linear. (c) Graph f together with the data.

The point B(1, 2) is one tenth of the way from the point A(3, 4) to the point P(x, y). Find x and y.

For what values of a is the distance between P(2, 3) and Q(6, a) greater than 5?

Find the standard equation of the circle with center (4, 5) that has an x-intercept at 0.

Find the other intercepts of the circle with center (4, 5) that has an x-intercept at 0.

A circle has center (4, 5). Find the slope-intercept form of the line tangent to the circle at the origin.

Find the slope-intercept form of the line that is perpendicular to the line and has x-intercept 4

A pizza costs $9.00 plus $0.80 for each topping. The sales tax rate is 10%. Find a function for the total cost T(x), where x is the number of toppings.

Find the domain of . Write your answer in interval notation.

Simplify the difference quotient for the function . Then, predict the difference quotient for based on your first answer.

A closed-top box has a square base of side y and height 2 feet. Express the surface area S of the box as a function of the volume V of the box.

If the point P(3, 2) is on the graph of a function f, find the corresponding point on the graph of

A salesman sells caps for $12. His commission is 10% on the first 1000 caps sold and 15% on any additional caps sold. Find a piecewise-defined function C that specifies the total commission if x caps are sold.

Find the standard equation of a parabola with vertical axis and vertex V(2, 1). What restriction must be made on the coefficient a if the graph of the parabola does not have any x-intercepts?

Find the minimum value of the parabola that has x-intercepts 2 and 4 and passes through the point (3, 15).

Find the mimimum value of the product p of two numbers, one of which is 9 less than twice the other.

A park offers group tours for 100 to 300 people at a time. For every person over 100, the cost for each ticket is reduced by one cent off the normal $4 per person charge. Find the number of people in the group that yields the maximum total cost for the group and also find the number of people in the group that yields the minimum total cost for the group.

Find the domain of if and .

A manufacturer produces 5 items per hour. The cost of producing y items is given by , where C is in thousands of dollars. Find a composite function C in terms of t hours and then use it to find the minimum cost.