# Precalculus with Limits

## Educators

PM
+ 3 more educators

Problem 1

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

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Problem 2

The point of intersection of the $x$ - and $y$ -axes is the_____, and the two axes divide the coordinate plane into four parts called _____.

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Problem 3

The _____ _____ is a result derived from the Pythagorean Theorem.

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Problem 4

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_____ ______.

Suzana M.

Problem 5

Plotting Points in the Cartesian Plane In
Exercises 5, plot the points in the Cartesian plane.
$$(-4,2),(-3,-6),(0,5),(1,-4),(0,0),(3,1)$$

Jimmy Y.

Problem 6

Plotting Points in the Cartesian Plane In
Exercises 6, plot the points in the Cartesian plane.
$$\left(1,-\frac{1}{3}\right),(0.5,-1),\left(\frac{3}{7}, 3\right),\left(-\frac{4}{3},-\frac{3}{7}\right),(-2,2.5)$$

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Problem 7

Finding the Coordinates of a Point In Exercises 7 and 8 , 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 8

Finding the Coordinates of a Point In Exercises 8, find the coordinates of the point. The point is on the $x$ -axis and 12 units to the left of the $y$ -axis.

Karuna R.

Problem 10

Determining Quadrant(s) for a Point, determine the quadrant(s) in which $(x, y)$ is
located so that the condition(s) is (are) satisfied.
$$x<0 \text { and } y<0$$

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Problem 11

Determining Quadrant(s) for a Point, determine the quadrant(s) in which $(x, y)$ is
located so that the condition(s) is (are) satisfied.
$$x=-4 \text { and } y>0$$

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Problem 12

Determining Quadrant(s) for a Point, 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 13

Determining Quadrant(s) for a Point, determine the quadrant(s) in which $(x, y)$ is
located so that the condition(s) is (are) satisfied.
$$x<0 \text { and }-y>0$$

PM
Pranati M.

Problem 14

Determining Quadrant(s) for a Point, determine the quadrant(s) in which $(x, y)$ is
located so that the condition(s) is (are) satisfied.
$$x y>0$$

Taylor S.

Problem 15

Sketching a Scatter Plot In Exercises 15 and $16,$ sketch a scatter plot of the data shown in the table. The table shows the number $y$ of Wal-Mart stores for each year $x$ from 2003 through $2010 .$ (Source: Wal-Mart Stores, Inc.)

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Problem 16

Sketching a Scatter Plot, sketch a scatter plot of the data shown in the table. 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 $)$

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Problem 17

Finding a Distance In Exercises $17-22,$ find the distance between the points.
$$(-2,6),(3,-6)$$

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Problem 18

Finding a Distance, find the distance between the points.
$$(8,5),(0,20)$$

AN
Aryan N.

Problem 19

Finding a Distance, find the distance between the points.
$$(1,4),(-5,-1)$$

YS
Yoochan S.

Problem 20

Finding a Distance, find the distance between the points.
$$(1,3),(3,-2)$$

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Problem 21

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

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Problem 22

Finding a Distance, find the distance between the points.
$$(9.5,-2.6),(-3.9,8.2)$$

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Problem 23

Verifying a Right Triangle In Exercises 23,
(a) find the length of each side of the right triangle, and
(b) show that these lengths satisfy the Pythagorean Theorem.

YS
Yitzchak S.

Problem 24

Verifying a Right Triangle In Exercises 24,
(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 25

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

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Problem 26

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

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Problem 27

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

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Problem 28

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

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Problem 29

Plotting, Distance, and Midpoint, (a) plot the points, (b) find the distance between
the points, and (c) find the midpoint of the line
segment joining the points.
$$(6,-3),(6,5)$$

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Problem 30

Plotting, Distance, and Midpoint, (a) plot the points, (b) find the distance between
the points, and (c) find the midpoint of the line
segment joining the points.
$$(1,4),(8,4)$$

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Problem 31

Plotting, Distance, and Midpoint, (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 32

Plotting, Distance, and Midpoint, (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 33

Plotting, Distance, and Midpoint, (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 34

Plotting, Distance, and Midpoint, (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 35

Plotting, Distance, and Midpoint, (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 36

Plotting, Distance, and Midpoint, (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 37

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 38

Sports A soccer player passes the ball from a point
that is 18 yards from the endline and 12 yards from the
sideline. A teammate who is 42 yards from the same
endline and 50 yards from the same sideline receives
the pass. (See figure.) How long is the pass?

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Problem 39

Sales The Coca-Cola Company had sales of $\$ 19,564$million in 2002 and$\$35,123$ million in $2010 .$ Use the
Midpoint Formula to estimate the sales in $2006 .$
Assume that the sales followed a linear pattern.
(Source: The Coca-Cola Company)

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Problem 40

Earnings per Share The earnings per share for
Big Lots, Inc. were $\$ 1.89$in 2008 and$\$2.83$ in $2010 .$
Use the Midpoint Formula to estimate the earnings
per share in $2009 .$ Assume that the earnings per share
followed a linear pattern. (Source: Big Lots, Inc.)

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Problem 41

Translating Points in the Plane In Exercises 41,
find the coordinates of the vertices of the polygon after
the indicated translation to a new position in the plane.

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Problem 42

Translating Points in the Plane In Exercises 42,
find the coordinates of the vertices of the polygon after
the indicated translation to a new position in the plane.

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Problem 43

Translating Points in the Plane,
find the coordinates of the vertices of the polygon after
the indicated translation to a new position in the plane.
Original coordinates of vertices: $(-7,-2),(-2,2),$
$(-2,-4),(-7,-4)$
Shift: eight units up, four units to the right

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Problem 44

Translating Points in the Plane,
find the coordinates of the vertices of the polygon after
the indicated translation to a new position in the plane.
Original coordinates of vertices: $(5,8),(3,6),(7,6)$
Shift: 6 units down, 10 units to the left

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Problem 45

Minimum Wage Use the graph below, which
shows the minimum wages in the United States (in
dollars) from 1950 through $2011 .$ (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 2011 .
(c) Use the percent increase from 1995 to 2011 to
predict the minimum wage in $2016 .$
(d) Do you believe that your prediction in part (c) is
reasonable? Explain.

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Problem 46

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.
$$\begin{array}{|c|c|c|c|c|c|c|c|c|}\hline x & {22} & {29} & {35} & {44} & {48} & {53} & {58} & {65} & {76} \\ \hline y & {53} & {74} & {57} & {66} & {79} & {90} & {76} & {93} & {99} \\ \hline\end{array}$$
(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 47

Using the Midpoint Formula 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 $y_{n m}$

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Problem 48

Using the Midpoint Formula Use the result of
Exercise 47 to find the coordinates of the endpoint of a
line segment when the coordinates of the other endpoint
and midpoint are, respectively,
$$\text { (a) }(1,-2),(4,-1) \text { and }\text { (b) }(-5,11),(2,4)$$

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Problem 49

Using the Midpoint Formula 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.

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Problem 50

Using the Midpoint Formula Use the result
of Exercise 49 to find the points that divide the line
segment joining the given points into four equal parts.
$$(\mathrm{a})(1,-2),(4,-1) \quad \text { (b) }(-2,-3),(0,0)$$

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Problem 51

Make a Conjecture Plot the points $(2,1),(-3,5),$
and $(7,-3)$ on a rectangular coordinate system. Then
change the signs of the indicated coordinates 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 52

Collinear Points Three or more points are
collinear when they all lie on the same line. Use the
steps following to determine whether 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.

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Problem 53

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.

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Problem 54

Think About It What is the $y$ -coordinate of any
point on the $x$ -axis? What is the $x$ -coordinate of any
point on the $y$ -axis?

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Problem 55

True or False? , determine whether
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 56

True or False? , determine whether
The points $(-8,4),(2,11),$ and $(-5,1)$ represent the
vertices of an isosceles triangle.

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Problem 57

True or False? , determine whether
If four points represent the vertices of a polygon, and
the four sides are equal, then the polygon must be a
square.

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Problem 58

HOW DOYOU SEE IT? 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. IThe plots are
labeled (i), (ii), (iii), and (iv).]
$$\begin{array}{ll}{\text { (a) }\left(x_{0}, y_{0}\right)} & {\text { (b) }\left(-2 x_{0}, y_{0}\right)} \\ {\text { (c) }\left(x_{0}, \frac{1}{2} y_{0}\right)} & {\text { (d) }\left(-x_{0},-y_{0}\right)}\end{array}$$

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Problem 59

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

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