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

Eric R.

Numerade Educator

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

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

$y<-5$

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

$x>4$

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

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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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Find the distance between the points.

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

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Find the distance between the points.

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

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(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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Show that the points form the vertices of the indicated polygon.

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

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Show that the points form the vertices of the indicated polygon.

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

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Show that the points form the vertices of the indicated polygon.

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

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Show that the points form the vertices of the indicated polygon.

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

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(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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(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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(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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(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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(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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(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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(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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(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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(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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(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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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.

Elizabeth S.

Numerade Educator

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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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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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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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.

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Determine whether the statement is true or false. Justify your answer.

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

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

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