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Mathematics LibreTexts

11.E: 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. (1, 3), (3, 1)
  2. (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.

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

  1. CNX_BMath_Figure_11_04_249_img.jpg
  2. CNX_BMath_Figure_11_04_250_img.jpg
  3. CNX_BMath_Figure_11_04_251_img.jpg
  4. CNX_BMath_Figure_11_04_252_img.jpg

Verify Solutions to an Equation in Two Variables

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

  1. 5x + y = 10
    1. (5, 1)
    2. (2, 0)
    3. (4, −10)
  2. 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.

  1. y = 4x − 1
x y (x, y)
0    
1    
-2    
  1. y = \(− \frac{1}{2}\)x + 3
x y (x, y)
0    
1    
-2    
  1. x + 2y = 5
x y (x, y)
  0  
1    
-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.

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

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

CNX_BMath_Figure_11_04_253_img.jpg

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

CNX_BMath_Figure_11_04_254_img.jpg

Graph a Linear Equation by Plotting Points

In the following exercises, graph by plotting points.

  1. y = 4x − 3
  2. y = −3x
  3. 2x + y = 7

Graph Vertical and Horizontal lines

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

  1. y = −2
  2. x = 3

11.3 - Graphing with Intercepts

Identify the Intercepts on a Graph

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

  1. CNX_BMath_Figure_11_04_261_img.jpg
  2. CNX_BMath_Figure_11_04_262_img.jpg

Find the Intercepts from an Equation of a Line

In the following exercises, find the intercepts.

  1. x + y = 5
  2. x − y = −1
  3. y = \(\frac{3}{4}\)x − 12
  4. y = 3x

Graph a Line Using the Intercepts

In the following exercises, graph using the intercepts.

  1.  −x + 3y = 3
  2. 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.

  1. x = 5
  2. y = −3
  3. 2x + y = 5
  4. x − y = 2
  5. y = \(\frac{1}{2}\)x + 2
  6. y = \(\frac{3}{4}\)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.

  1. CNX_BMath_Figure_11_04_265_img.jpg
  2. CNX_BMath_Figure_11_04_266_img.jpg
  3. CNX_BMath_Figure_11_04_267_img.jpg
  4. CNX_BMath_Figure_11_04_268_img.jpg

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

  1. \(\frac{1}{3}\) 
  2. \(\frac{3}{2}\) 
  3. \(− \frac{2}{3}\) 
  4. \(− \frac{1}{2}\)

Find the Slope of a Line from its Graph

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

  1. CNX_BMath_Figure_11_04_273_img.jpg
  2. CNX_BMath_Figure_11_04_274_img.jpg
  3. CNX_BMath_Figure_11_04_275_img.jpg
  4. CNX_BMath_Figure_11_04_276_img.jpg

Find the Slope of Horizontal and Vertical Lines

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

  1. y = 2
  2. x = 5
  3. x = −3
  4. 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.

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

  1. (2, −2); m = \(\frac{5}{2}\) 
  2. (−3, 4); m = \(− \frac{1}{3}\)

Solve Slope Applications

In the following exercise, solve the slope application.

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

PRACTICE TEST

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

CNX_BMath_Figure_11_04_280_img.jpg

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

CNX_BMath_Figure_11_04_281_img.jpg

  1. Find the x-intercept and y-intercept of the equation 3x − y = 6.
  2. Is (1, 3) a solution to the equation x + 4y = 12? How do you know?
  3. Complete the table to find four solutions to the equation y = − x + 1.
x y (x, y)
0    
1    
3    
-2    
  1. 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.

  1. y = −3x
  2. 2x + 3y = −6

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

  1. CNX_BMath_Figure_11_04_284_img.jpg
  2. CNX_BMath_Figure_11_04_285_img.jpg
  3. Use the slope formula to find the slope of the line between (0, −4) and (5, 2).
  4. Find the slope of the line y = 2.
  5. Graph the line passing through (1, 1) with slope m = \(\frac{3}{2}\).
  6. A bicycle route climbs 20 feet for 1,000 feet of horizontal distance. What is the slope of the route?

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