# Precalculus with Limits (2010)

## Educators

ER
ES

### Problem 1

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

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### Problem 2

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

ER
Eric R.

### Problem 3

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 ________ ________.

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### Problem 4

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

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### Problem 5

Approximate the coordinates of the points.

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### Problem 6

Approximate the coordinates of the points.

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### Problem 7

Plot the points in the Cartesian plane.
$(-4,2),(-3,-6),(0,5),(1,-4)$

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### Problem 8

Plot the points in the Cartesian plane.
$(0,0),(3,1),(-2,4),(1,-1)$

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### Problem 9

Plot the points in the Cartesian plane.
$(3,8),(0.5,-1),(5,-6),(-2,2.5)$

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### Problem 10

Plot the points in the Cartesian plane.
$\left(1,-\frac{1}{3}\right),\left(\frac{3}{4}, 3\right),(-3,4),\left(-\frac{4}{3},-\frac{3}{2}\right)$

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### Problem 11

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.

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### Problem 12

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.

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### Problem 13

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

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### Problem 14

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

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### Problem 15

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x>0$ and $y<0$

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### Problem 16

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x<0$ and $y<0$

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### Problem 17

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x=-4$ and $y>0$

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### Problem 18

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x>2$ and $y=3$

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### Problem 19

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$y<-5$

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### Problem 20

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x>4$

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### Problem 21

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x<0$ and $-y>0$

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### Problem 22

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$-x>0$ and $y<0$

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### Problem 23

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x y>0$

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### Problem 24

Determine the quadrant(s) in which $(x, y)$ is located so that the condition(s) is (are) satisfied.
$x y<0$

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### Problem 25

The table shows the number of Wal-Mart stores for each year from 2000 through 2007.

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### Problem 26

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.

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### Problem 27

Find the distance between the points.
$(6,-3),(6,5)$

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### Problem 28

Find the distance between the points.
$(1,4),(8,4)$

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### Problem 29

Find the distance between the points.
$(-3,-1),(2,-1)$

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### Problem 30

Find the distance between the points.
$(-3,-4),(-3,6)$

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### Problem 31

Find the distance between the points.
$(-2,6),(3,-6)$

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### Problem 32

Find the distance between the points.
$(8,5),(0,20)$

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### Problem 33

Find the distance between the points.
$(1,4),(-5,-1)$

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### Problem 34

Find the distance between the points.
$(1,3),(3,-2)$

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### Problem 35

Find the distance between the points.
$\left(\frac{1}{2}, \frac{4}{3}\right),(2,-1)$

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### Problem 36

Find the distance between the points.
$\left(-\frac{2}{3}, 3\right),\left(-1, \frac{5}{4}\right)$

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### Problem 37

Find the distance between the points.
$(-4.2,3.1),(-12.5,4.8)$

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### Problem 38

Find the distance between the points.
$(9.5,-2.6),(-3.9,8.2)$

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### Problem 39

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

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### Problem 40

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

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### Problem 41

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

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### Problem 42

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

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### Problem 43

Show that the points form the vertices of the indicated polygon.
Right triangle: $(4,0),(2,1),(-1,-5)$

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### Problem 44

Show that the points form the vertices of the indicated polygon.
Right triangle: $(-1,3),(3,5),(5,1)$

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### Problem 45

Show that the points form the vertices of the indicated polygon.
Isosceles triangle: $(1,-3),(3,2),(-2,4)$

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### Problem 46

Show that the points form the vertices of the indicated polygon.
Isosceles triangle: $(2,3),(4,9),(-2,7)$

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### Problem 47

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(1,1),(9,7)$

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### Problem 48

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(1,12),(6,0)$

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### Problem 49

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(-4,10),(4,-5)$

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### Problem 50

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(-7,-4),(2,8)$

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### Problem 51

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(-1,2),(5,4)$

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### Problem 52

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(2,10),(10,2)$

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### Problem 53

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$\left(\frac{1}{2}, 1\right),\left(-\frac{5}{2}, \frac{4}{3}\right)$

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### Problem 54

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$\left(-\frac{1}{3},-\frac{1}{3}\right),\left(-\frac{1}{6},-\frac{1}{2}\right)$

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### Problem 55

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(6.2,5.4),(-3.7,1.8)$

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### Problem 56

(a) plot the points, (b) find the distance between the points, and (c) find the midpoint of the line segment joining the points.
$(-16.8,12.3),(5.6,4.9)$

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### Problem 57

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?

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### Problem 58

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?

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### Problem 59

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.
Big Lots
$\begin{array}{r|r}{2003} & {\$ 4174} \\ {2007} & {\$4656}\end{array}$

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### Problem 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.
Dollar Tree
$\begin{array}{r|r}{2003} & {\$ 2800} \\ {2007} & {\$4243}\end{array}$

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### Problem 61

The polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

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### Problem 62

The polygon is shifted to a new position in the plane. Find the coordinates of the vertices of the polygon in its new position.

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### Problem 63

Original coordinates of vertices: $(-7,-2),(-2,2),$
$(-2,-4),(-7,-4)$
Shift: eight units upward, four units to the right

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### Problem 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

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### Problem 65

Use the graph, which shows the average retail prices of 1 gallon of whole milk from 1996 to 2007.
Approximate the highest price of a gallon of whole milk shown in the graph. When did this occur?

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### Problem 66

Use the graph, which shows the average retail prices of 1 gallon of whole milk from 1996 to 2007.
Approximate the percent change in the price of milk from the price in 1996 to the highest price shown in the graph.

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### Problem 67

The graph shows the average costs of a 30-second television spot (in thousands of dollars) during the Super Bowl from 2000 to 2008.
(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 .

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### Problem 68

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

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### Problem 69

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.

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### Problem 70

Use the graph below, which shows the minimum wage in the United States (in dollars) from 1950 to 2009.
(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.

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### Problem 71

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.

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### Problem 72

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.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline x & {22} & {29} & {35} & {40} & {44} & {48} & {53} & {58} & {65} & {76} \\ \hline y & {53} & {74} & {57} & {66} & {79} & {90} & {76} & {93} & {83} & {99} \\ \hline\end{array}$$
(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.

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### Problem 73

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.
(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?

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### Problem 74

The table shows the numbers of men's $M$ and women's $W$ college basketball teams for each year $x$ from 1994 through 2007.
(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?

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### Problem 75

A line segment has $\left(x_{1}, y_{1}\right)$ as one endpoint and $\left(x_{m}, y_{m}\right)$ as its midpoint. Find the other endpoint $\left(x_{2}, y_{2}\right)$ of the line segment in terms of $x_{1}, y_{1}, x_{m},$ and $y_{m}$ .

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### 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 (b) $(-5,11),(2,4)$ .

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### Problem 77

Use the Midpoint Formula three times to find the three points that divide the line segment joining $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$ into four parts.

ES
Elizabeth S.

### 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) $(1,-2),(4,-1) \quad$ (b) $(-2,-3),(0,0)$

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### Problem 79

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.

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### Problem 80

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 to from to and from to 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.

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### Problem 81

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

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### Problem 82

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

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### Problem 83

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.

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### Problem 84

CAPSTONE Use the plot of the point $\left(x_{0}, y_{0}\right)$ 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) $\left(x_{0},-y_{0}\right) \quad$ (b) $\left(-2 x_{0}, y_{0}\right)$
(c) $\left(x_{0}, \frac{1}{2} y_{0}\right) \quad$ (d) $\left(-x_{0},-y_{0}\right)$

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### Problem 85

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

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