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Chapter 1

Functions and Their Graphs

Educators


Problem 1

Match each term with its definition.
(a) $ x $-axis (i) point of intersection of vertical axis and horizontal axis
(b) $ y $-axis (ii) directed distance from the $ x $-axis
(c) origin (iii) directed distance from the $ y $-axis
(d) quadrants (iv) four regions of the coordinate plane
(e) $ x $-coordinate (v) horizontal real number line
(f) $ y $-coordinate (vi) vertical real number line

Heather Z.
Numerade Educator

Problem 2

In Exercises 2 - 4, fill in the blanks.

An ordered pair of real numbers can be represented in a plane called the rectangular coordinate system or the ______________ plane.

Heather Z.
Numerade Educator

Problem 3

In Exercises 2 - 4, fill in the blanks.

The __________ _____________ is a result derived from the Pythagorean Theorem.

Heather Z.
Numerade Educator

Problem 4

In Exercises 2 - 4, fill in the blanks.

Finding the average values of the representative coordinates of the two endpoints of a line segment in a coordinate plane is also known as using the _____________ _______________.

Heather Z.
Numerade Educator

Problem 5

In Exercises 5 and 6, approximate the coordinates of the points.

Heather Z.
Numerade Educator

Problem 6

In Exercises 5 and 6, approximate the coordinates of the points.

Heather Z.
Numerade Educator

Problem 7

In Exercises 7-10, plot the points in the Cartesian plane.

$ (-4, 2) $, $ (-3, -6) $, $ (0, 5) $, $ (1, -4) $

Heather Z.
Numerade Educator

Problem 8

In Exercises 7-10, plot the points in the Cartesian plane.

$ (0, 0) $, $ (3, 1) $, $ (-2, 4) $, $ (1, -1) $

Heather Z.
Numerade Educator

Problem 9

In Exercises 7-10, plot the points in the Cartesian plane.

$ (3, 8) $, $ (0.5, -1) $, $ (5, -6) $, $ (-2, 2.5) $

Heather Z.
Numerade Educator

Problem 10

In Exercises 7-10, plot the points in the Cartesian plane.

$ (1, -\frac{1}{3}) $, $ (\frac{3}{4}, 3) $, $ (-3, 4) $, $ (-\frac{4}{3}, -\frac{3}{2}) $

Heather Z.
Numerade Educator

Problem 11

In Exercises 11-14, find the coordinates of the point.

The point is located three units to the left of the $ y $-axis and four units above the $ x $-axis.

Heather Z.
Numerade Educator

Problem 12

In Exercises 11-14, find the coordinates of the point.

The point is located eight units below the $ x $-axis and four units to the right of the $ y $-axis.

Heather Z.
Numerade Educator

Problem 13

In Exercises 11-14, find the coordinates of the point.

The point is located five units below the $ x $-axis and and the coordinates of the point are equal.

Heather Z.
Numerade Educator

Problem 14

In Exercises 11-14, find the coordinates of the point.

The point is on the $ x $-axis and 12 units to the left of the $ y $-axis.

Heather Z.
Numerade Educator

Problem 15

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x > 0 $ and $ y < 0 $

Heather Z.
Numerade Educator

Problem 16

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x < 0 $ and $ y < 0 $

Heather Z.
Numerade Educator

Problem 17

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x = -4 $ and $ y > 0 $

Heather Z.
Numerade Educator

Problem 18

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x > 2 $ and $ y = 3 $

Heather Z.
Numerade Educator

Problem 19

In Exercises $15-24,$ determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$$
y<-5
$$

Heather Z.
Numerade Educator

Problem 20

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x > 4 $

Heather Z.
Numerade Educator

Problem 21

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ x < 0 $ and $ -y > 0 $

Yuki H.
Numerade Educator

Problem 22

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ -x > 0 $ and $ y < 0 $

Heather Z.
Numerade Educator

Problem 23

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ xy > 0 $

Heather Z.
Numerade Educator

Problem 24

In Exercise 15-24, determine the quadrant(s) in which $ (x, y) $ is located so that the condition(s) is (are) satisfied.

$ xy < 0 $

Heather Z.
Numerade Educator

Problem 25

In Exercises 25 and 26, sketch a scatter plot of the data shown in the table.

NUMBER OF STORES The table shows the number $ y $ of Wal-Mart stores for each year $ x $ from 2000 through 2007. (Source: Wal-Mart Stores, Inc.)

Heather Z.
Numerade Educator

Problem 26

In Exercises 25 and 26, sketch a scatter plot of the data shown in the table.

METEOROLOGY The table shows the lowest temperature on record $ y $ (in degrees Fahrenheit) in Duluth, Minnesota for each month $ x $, where $ x = 1 $ represents January. (Source: NOAA)

Heather Z.
Numerade Educator

Problem 27

In Exercises 27-38, find the distance between the points.

$ (6, -3) $, $ (6, 5) $

Heather Z.
Numerade Educator

Problem 28

In Exercises 27-38, find the distance between the points.

$ (1, 4) $, $ (8, 4) $

Heather Z.
Numerade Educator

Problem 29

In Exercises 27-38, find the distance between the points.

$ (-3, -1) $, $ (2, -1) $

Heather Z.
Numerade Educator

Problem 30

In Exercises 27-38, find the distance between the points.

$ (-3, -4) $, $ (-3, 6) $

Heather Z.
Numerade Educator

Problem 31

In Exercises 27-38, find the distance between the points.

$ (-2, 6) $, $ (3, -6) $

Heather Z.
Numerade Educator

Problem 32

In Exercises 27-38, find the distance between the points.

$ (8, 5) $, $ (0, 20) $

Heather Z.
Numerade Educator

Problem 33

In Exercises 27-38, find the distance between the points.

$ (1, 4) $, $ (-5, -1) $

Heather Z.
Numerade Educator

Problem 34

In Exercises 27-38, find the distance between the points.

$ (1, 3) $, $ (3, -2) $

Heather Z.
Numerade Educator

Problem 35

In Exercises 27-38, find the distance between the points.

$ (\frac{1}{2}, \frac{4}{3}) $, $ (2, -1) $

Heather Z.
Numerade Educator

Problem 36

In Exercises 27-38, find the distance between the points.

$ (-\frac{2}{3}, 3) $, $ (-1, \frac{5}{4}) $

Heather Z.
Numerade Educator

Problem 37

In Exercises 27-38, find the distance between the points.

$ (-4.2, 3.1) $, $ (-12.5, 4.8) $

Yuki H.
Numerade Educator

Problem 38

In Exercises 27-38, find the distance between the points.

$ (9.5, -2.6) $, $ (-3.9, 8.2) $

Heather Z.
Numerade Educator

Problem 39

In Exercises 39-42, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem.

Heather Z.
Numerade Educator

Problem 40

In Exercises 39-42, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem.

Heather Z.
Numerade Educator

Problem 41

In Exercises 39-42, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem.

Heather Z.
Numerade Educator

Problem 42

In Exercises 39-42, (a) find the length of each side of the right triangle, and (b) show that these lengths satisfy the Pythagorean Theorem.

Heather Z.
Numerade Educator

Problem 43

In Exercises 43-46, show that the points form the vertices of the indicated polygon.

Right triangle: $ (4, 0) $, $ (2, 1) $, $ (-1, -5) $

Heather Z.
Numerade Educator

Problem 44

In Exercises 43-46, show that the points form the vertices of the indicated polygon.

Right triangle: $ (-1, 3) $, $ (3, 5) $, $ (5, 1) $

Heather Z.
Numerade Educator

Problem 45

In Exercises 43-46, show that the points form the vertices of the indicated polygon.

Isosceles triangle: $ (1, -3) $, $ (3, 2) $, $ (-2, 4) $

Heather Z.
Numerade Educator

Problem 46

In Exercises 43-46, show that the points form the vertices of the indicated polygon.

Isosceles triangle: $ (2, 3) $, $ (4, 9) $, $ (-2, 7) $

Heather Z.
Numerade Educator

Problem 47

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (1, 1) $, $ (9, 7) $

Heather Z.
Numerade Educator

Problem 48

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (1, 12) $, $ (6, 0) $

Heather Z.
Numerade Educator

Problem 49

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (-4, 10) $, $ (4, -5) $

Heather Z.
Numerade Educator

Problem 50

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (-7, -4) $, $ (2, 8) $

Heather Z.
Numerade Educator

Problem 51

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (-1, 2) $, $ (5, 4) $

Heather Z.
Numerade Educator

Problem 52

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (2, 10) $, $ (10, 2) $

Heather Z.
Numerade Educator

Problem 53

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (\frac{1}{2}, 1) $, $ (-\frac{5}{2}, \frac{4}{3}) $

Heather Z.
Numerade Educator

Problem 54

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (-\frac{1}{3}, -\frac{1}{3}) $, $ (-\frac{1}{6}, -\frac{1}{2}) $

Heather Z.
Numerade Educator

Problem 55

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (6.2, 5.4) $, $ (-3.7, 1.8) $

Heather Z.
Numerade Educator

Problem 56

In Exercises 47-56, (a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.

$ (-16.8, 12.3) $, $ (5.6, 4.9) $

Heather Z.
Numerade Educator

Problem 57

FLYING DISTANCE An airplane flies from Naples, Italy in a straight line to Rome, Italy, which is 120 kilometers north and 150 kilometers west of Naples. How far does the plane fly?

Heather Z.
Numerade Educator

Problem 58

SPORTS A soccer player passes the ball from a point that is 18 yards from the endline and 12 yards from the sideline. The pass is received by a teammate who is 42 yards from the same endline and 50 yards from the same sideline, as shown in the figure. How long is the pass?

Heather Z.
Numerade Educator

Problem 59

SALES In Exercises 59 and 60, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc.in 2005, given the sales in 2003 and 2007. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.;Dollar Tree Stores, Inc.)

Big Lots

Heather Z.
Numerade Educator

Problem 60

SALES In Exercises 59 and 60, use the Midpoint Formula to estimate the sales of Big Lots, Inc. and Dollar Tree Stores, Inc.in 2005, given the sales in 2003 and 2007. Assume that the sales followed a linear pattern. (Source: Big Lots, Inc.;Dollar Tree Stores, Inc.)

Dollar Tree

Heather Z.
Numerade Educator

Problem 61

In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

Heather Z.
Numerade Educator

Problem 62

In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

Heather Z.
Numerade Educator

Problem 63

In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

Original coordinates of vertices: $ (-7, -2) $, $ (-2, 2) $, $ (-2, -4) $, $ (-7, -4) $

Shift: eight units upward, four units to the right

Heather Z.
Numerade Educator

Problem 64

In Exercises 61-64, the polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

Original coordinates of vertices: $ (5, 8) $, $ (3, 6) $, $ (7, 6) $, $ (5, 2) $

Shift: 6 units downward, 10 units to the left

Heather Z.
Numerade Educator

Problem 65

RETAIL PRICE In Exercises 65 and 66, use the graph,which shows the average retail prices of 1 gallon of whole milk from 1996 to 2007. (Source: U.S. Bureau of Labor Statistics)

Approximate the highest price of a gallon of whole milk shown in the graph. When did this occur?

Heather Z.
Numerade Educator

Problem 66

RETAIL PRICE In Exercises 65 and 66, use the graph,which shows the average retail prices of 1 gallon of whole milk from 1996 to 2007. (Source: U.S. Bureau of Labor Statistics)

Approximate the percent change in the price of milk from the price in 1996 to the highest price shown in the graph.

Heather Z.
Numerade Educator

Problem 67

ADVERTISING The graph shows the average costs of a 30-second television spot (in thousands of dollars) during the Super Bowl from 2000 to 2008. (Source: Nielson Media and TNS Media Intelligence)

FIGURE FOR 67

(a) Estimate the percent increase in the average cost of a 30-second spot from Super Bowl XXXIV in 2000 to Super Bowl XXXVIII in 2004.

(b) Estimate the percent increase in the average cost of a 30-second spot from Super Bowl XXXIV in 2000 to Super Bowl XLII in 2008.

Heather Z.
Numerade Educator

Problem 68

ADVERTISING The graph shows the average costs of a 30-second television spot (in thousands of dollars) during the Academy Awards from 1995 to 2007. (Source: Nielson Monitor-Plus)

(a) Estimate the percent increase in the average cost of a 30-second spot in 1996 to the cost in 2002.

(b) Estimate the percent increase in the average cost of a 30-second spot in 1996 to the cost in 2007.

Heather Z.
Numerade Educator

Problem 69

MUSIC The graph shows the numbers of performers who were elected to the Rock and Roll Hall of Fame from 1991 through 2008. Describe any trends in the data. From these trends, predict the number of performers elected in 2010. (Source: rockhall.com)

Heather Z.
Numerade Educator

Problem 70

LABOR FORCE Use the graph below, which shows the minimum wage in the United States (in dollars) from 1950 to 2009. (Source: U.S. Department of Labor)

(a) Which decade shows the greatest increase in minimum wage?

(b) Approximate the percent increases in the minimum wage from 1990 to 1995 and from 1995 to 2009.

(c) Use the percent increase from 1995 to 2009 to predict the minimum wage in 2013.

(d) Do you believe that your prediction in part (c) is reasonable? Explain.

Heather Z.
Numerade Educator

Problem 71

SALES The Coca-Cola Company had sales of $\$19,805$ million in 1999 and $\$28,857$ million in 2007. Use the Midpoint Formula to estimate the sales in 2003. Assume that the sales followed a linear pattern.(Source: The Coca-Cola Company)

Heather Z.
Numerade Educator

Problem 72

DATA ANALYSIS: EXAM SCORES The table shows the mathematics entrance test scores $ x $ and the final examination scores $ y $ in an algebra course for a sample of 10 students.

(a) Sketch a scatter plot of the data.

(b) Find the entrance test score of any student with a final exam score in the 80s.

(c) Does a higher entrance test score imply a higher final exam score? Explain.

Heather Z.
Numerade Educator

Problem 73

DATA ANALYSIS: MAIL The table shows the number $ y $ of pieces of mail handled (in billions) by the U.S. Postal Service for each year $ x $ from 1996 through 2008. (Source: U.S. Postal Service)

TABLE FOR 73

(a) Sketch a scatter plot of the data.

(b) Approximate the year in which there was the greatest decrease in the number of pieces of mail handled.

(c) Why do you think the number of pieces of mail handled decreased?

Heather Z.
Numerade Educator

Problem 74

DATA ANALYSIS: ATHLETICS

The table shows the numbers of men's $ M $ and women's $ W $ college basketball teams for each year $ x $ from 1994 through 2007. (Source: National Collegiate Athletic Association)

(a) Sketch scatter plots of these two sets of data on the same set of coordinate axes.

(b) Find the year in which the numbers of men’s and women’s teams were nearly equal.

(c) Find the year in which the difference between the numbers of men’s and women’s teams was the greatest. What was this difference?

Heather Z.
Numerade Educator

Problem 75

A line segment has $ (x_1, y_1) $ as one endpoint and $ (x_m, y_m) $ as its midpoint. Find the other endpoint $ (x_2, y_2) $ of the line segment in terms of $ x_1, y_1, x_m, $ and $ y_m $.

Heather Z.
Numerade Educator

Problem 76

Use the result of Exercise 75 to find the coordinates of the endpoint of a line segment if the coordinates of the other endpoint and midpoint are, respectively, (a) $ (1, -2) $, $ (4, -1) $ and (a) $ (-5, 11) $, $ (2, 4) $.

Heather Z.
Numerade Educator

Problem 77

Use the Midpoint Formula three times to find the three points that divide the line segment joining $ (x_1, y_1) $ and $ (x_2, y_2) $ into four parts.

Heather Z.
Numerade Educator

Problem 78

Use the result of Exercise 77 to find the points that divide the line segment joining the given points into four equal parts.

(a) $ (1, -2) $, $ (4, -1) $ and (b) $ (-2, -3) $, $ (0, 0) $.

Heather Z.
Numerade Educator

Problem 79

MAKE A CONJECTURE Plot the points $ (2, 1) $, $ (-3, 5) $ and $ (7, -3) $ on a rectangular coordinate system. Then change the sign of the $ x $-coordinate of each point and plot the three new points on the same rectangular coordinate system. Make a conjecture about the location of a point when each of the following occurs.

(a) The sign of the $ x $-coordinate is changed.

(b) The sign of the $ y $-coordinate is changed.

(c) The signs of both the $ x $- and $ y $-coordinates are changed.

Heather Z.
Numerade Educator

Problem 80

COLLINEAR POINTS Three or more points are collinear if they all lie on the same line. Use the steps below to determine if the set of points {$ A(2, 3) $, $ B(2, 6) $, $ C(6, 3) $} and the set of points {$ A(8, 3) $, $ B(5, 2) $, $ C(2, 1) $} are collinear.

(a) For each set of points, use the Distance Formula to find the distances from $ A $ to $ B $ from $ B $ to $ C $ and from $ A $ to $ C $. What relationship exists among these distances for each set of points?

(b) Plot each set of points in the Cartesian plane. Do all the points of either set appear to lie on the same line?

(c) Compare your conclusions from part (a) with the conclusions you made from the graphs in part (b). Make a general statement about how to use the Distance Formula to determine collinearity.

Heather Z.
Numerade Educator

Problem 81

TRUE OR FALSE? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer.

In order to divide a line segment into 16 equal parts, you would have to use the Midpoint Formula 16 times.

Heather Z.
Numerade Educator

Problem 82

TRUE OR FALSE? In Exercises 81 and 82, determine whether the statement is true or false. Justify your answer.

The points and $ (-8, 4) $, $ (2, 11) $ and $ (-5, 1) $ represent the vertices of an isosceles triangle.

Heather Z.
Numerade Educator

Problem 83

THINK ABOUT IT When plotting points on the rectangular coordinate system, is it true that the scales on the $ x $- and $ y $-axes must be the same? Explain.

Heather Z.
Numerade Educator

Problem 84

CAPSTONE Use the plot of the point $ (x_0, y_0) $ in the figure. Match the transformation of the point with the correct plot. Explain your reasoning. [The plots are labeled (i), (ii), (iii), and (iv).]

(a) $ (x_0, - y_0) $ (b) $ (- 2x_0, y_0) $
(c) $ (x_0, \frac{1}{2}y_0) $ (d) $ (- x_0, - y_0) $

Heather Z.
Numerade Educator

Problem 85

PROOF Prove that the diagonals of the parallelogram in the figure intersect at their midpoints.

Heather Z.
Numerade Educator