11.9: Graphs (Exercises)

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11.1 - Use the Rectangular Coordinate System

Plot Points in a Rectangular Coordinate System

In the following exercises, plot each point in a rectangular coordinate system.

(1, 3), (3, 1)
(2, 5), (5, 2)

In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.

(a) (−1, −5) (b) (−3, 4) (c) (2, −3) (d) $\left(1, \dfrac{5}{2}\right)$
(a) (3, −2) (b) (−4, −1) (c) (−5, 4) (d) $\left(2, \dfrac{10}{3}\right)$

Identify Points on a Graph

In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.

$The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 5, 3, “b” at 2, -1, “c” at -3,-2, and “d” at -1,4.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at -2, 2, “b” at 3, 5, “c” at 4,-1, and “d” at -1,3.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 2, 0, “b” at 0, -5, “c” at -4,0, and “d” at 0,3.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. “a” is plotted at 0, 4, “b” at 5, 0, “c” at 0,-1, and “d” at -3,0.$

Verify Solutions to an Equation in Two Variables

In the following exercises, find the ordered pairs that are solutions to the given equation.

5x + y = 10
1. (5, 1)
2. (2, 0)
3. (4, −10)
y = 6x − 2
1. (1, 4)
2. $\left(\dfrac{1}{3} , 0\right)$
3. (6, −2)

Complete a Table of Solutions to a Linear Equation in Two Variables

In the following exercises, complete the table to find solutions to each linear equation.

y = 4x − 1

x	y	(x, y)
0
1
-2

y = $− \dfrac{1}{2}$x + 3

x	y	(x, y)
0
1
-2

x + 2y = 5

x	y	(x, y)
	0
1
-1

3x − 2y = 6

x	y	(x, y)
0
	0
-2

Find Solutions to a Linear Equation in Two Variables

In the following exercises, find three solutions to each linear equation.

x + y = 3
x + y = −4
y = 3x + 1
y = − x − 1

11.2 - Graphing Linear Equations

Recognize the Relation Between the Solutions of an Equation and its Graph

In the following exercises, for each ordered pair, decide (a) if the ordered pair is a solution to the equation. (b) if the point is on the line.

y = − x + 4
1. (0, 4)
2. (−1, 3)
3. (2, 2)
4. (−2, 6)

$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair 4, 0”.$

y = $\dfrac{2}{3}$x − 1
1. (0, −1)
2. (3, 1)
3. (−3, −3)
4. (6, 4)

$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, -1” and “ordered pair 3, 1”.$

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

y = 4x − 3
y = −3x
2x + y = 7

Graph Vertical and Horizontal lines

In the following exercises, graph the vertical or horizontal lines.

y = −2
x = 3

11.3 - Graphing with Intercepts

Identify the Intercepts on a Graph

In the following exercises, find the x- and y-intercepts.

$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.$
$The graph shows the x y-coordinate plane. The x-axis runs from -1 to 6. The y-axis runs from -4 to 2. A line passes through the points “ordered pair 5, 1” and “ordered pair 0, -3”.$

Find the Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

x + y = 5
x − y = −1
y = $\dfrac{3}{4}$x − 12
y = 3x

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

−x + 3y = 3
x + y = −2

Choose the Most Convenient Method to Graph a Line

In the following exercises, identify the most convenient method to graph each line.

x = 5
y = −3
2x + y = 5
x − y = 2
y = $\dfrac{1}{2}$x + 2
y = $\dfrac{3}{4}$x − 1

11.4 - Understand Slope of a Line

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

$The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 4 and the point in column 4 row 2.$
$The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 5 and the point in column 4 row 1.$
$The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 3 and the point in column 4 row 4.$
$The figure shows a grid of evenly spaced dots. There are 5 rows and 5 columns. There is a rubber band style loop connecting the point in column 1 row 2 and the point in column 4 row 4.$

In the following exercises, model each slope. Draw a picture to show your results.

$\dfrac{1}{3}$
$\dfrac{3}{2}$
$− \dfrac{2}{3}$
$− \dfrac{1}{2}$

Find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 0” and “ordered pair 2, -6”.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair 0, 4” and “ordered pair -4, 0”.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -4, -4” and “ordered pair 5, -1”.$
$The graph shows the x y-coordinate plane. The axes run from -7 to 7. A line passes through the points “ordered pair -3, 6” and “ordered pair 5, 2”.$

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

y = 2
x = 5
x = −3
y = −1

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

(2, 1), (4, 5)
(−1, −1), (0, −5)
(3, 5), (4, −1)
(−5, −2), (3, 2)

Graph a Line Given a Point and the Slope

In the following exercises, graph the line given a point and the slope.

(2, −2); m = $\dfrac{5}{2}$
(−3, 4); m = $− \dfrac{1}{3}$

Solve Slope Applications

In the following exercise, solve the slope application.

A roof has rise 10 feet and run 15 feet. What is its slope?

PRACTICE TEST

Plot and label these points:
1. (2, 5)
2. (−1, −3)
3. (−4, 0)
4. (3, −5)
5. (−2, 1)
Name the ordered pair for each point shown.

$The graph shows the x y-coordinate plane. The axes extend from -7 to 7. A is plotted at -4, 1, B at 3, 2, C at 0, -2, D at -1, -4, and E at 4,-3.$

Find the x-intercept and y-intercept on the line shown.

$The graph shows the x y-coordinate plane. The x-axis runs from -7 to 7. The y-axis runs from -7 to 7. A line passes through the points “ordered pair 4, 0” and “ordered pair 0, -2”.$

Find the x-intercept and y-intercept of the equation 3x − y = 6.
Is (1, 3) a solution to the equation x + 4y = 12? How do you know?
Complete the table to find four solutions to the equation y = − x + 1.

x	y	(x, y)
0
1
3
-2

Complete the table to find three solutions to the equation 4x + y = 8.

x	y	(x, y)
0
	0
3

In the following exercises, find three solutions to each equation and then graph each line.

y = −3x
2x + 3y = −6