Algebra 2 by Richard Wright

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0-05 Graph Equations of Lines

Mr. Wright teaches the lesson.

Objectives:

SDA NAD Content Standards (2018): AII.4.1, AII.5.3, AII.7.1

stock trading monitor
Figure 1: Stock trading monitor (Pixabay/3844328)

They say a picture is worth a thousand words. Many things can be read from a graph of a function such as solutions to equations, values at different points, and general trends. That is the reason things like stock prices are usually represented as graphs. This lesson is about making graphs of linear equations.

Slope-Intercept Form

Linear equations are equations that graph a straight line. They can be written in the form y = mx + b where m is the slope and b is the y-intercept. The y-intercept is the place where the graph of the line crosses the y-axis. The equation y = mx + b is called slope-intercept form.

y-intercept
Figure 2: Line with the y-intercept

To graph any equation, you can always make a table of values by choosing a range of x values and plugging those into the equation to find the y values. Then plot the points and draw a line.

Graph by Making a Table

To graph a function by making a table,

  1. Choose a reasonable range of x values usually including negatives.
  2. Substitute each x value into the function to find the corresponding y value.
  3. Plot the points on a coordinate plane.
  4. Draw the line through the points.

Example 1

Graph \(y=\frac{1}{2}x-2\) by making a table.

Solution

Choose a reasonable range of x values such as −4 to 4. Take each of those number and substitute it for x in the equation to find y.

x−4−3−2−101234
y−4−3.5−3−2.5−2−1.5−1−0.50

Plot the points and draw a line.

y=1/2 x-2
Figure 3: \(y=\frac{1}{2}x-2\)
Graph using Slope-Intercept Form

To graph linear equations in slope-intercept form, y = mx + b

  1. Solve equation for y if needed.
  2. Plot the y-intercept.
  3. From there move up and over the slope to find another couple of points.
  4. Draw a line neatly through the points.

Example 2

Graph y = 2x + 1.

Solution

By comparing to y = mx + b, we can see that the slope, m, is 2 and the y-intercept, b, is 1. Plot a point at 1 on the y-axis. Then from there follow the slope of 2, or \(\frac{2}{1}\), up 2 and over 1 to plot a second point. Repeat this to get several more points. Finally draw a straight line through the points.

y=2x+1
Figure 4: y = 2x + 1

Example 3

Graph y = −x − 3.

Solution

By comparing to y = mx + b we can see that the slope, m, is −1 and the y-intercept, b, is −3. Plot a point at −3 on the y-axis. Then from there follow the slope of −1, or \(\frac{-1}{1}\), down 1 and over 1 to plot a second point. If needed, the slope of −1 could also be \(\frac{1}{-1}\), which is up 1 and left 1 to get additional points. Finally draw a straight line through the points.

y=-x-3
Figure 5: y = −x − 3

Standard Form

The standard form of linear equations is Ax + By = C where A, B, and C are integers and A is usually positive.

Graph Linear Equations in Standard Form

To graph linear equations in standard form, Ax + By = C

  1. Find the x- and y-intercepts by letting the other variable = 0.
    1. At the x-intercept, the y-coordinate is 0, (x, 0).
      1. Ax + B(0) = C
      2. Ax = C
      3. \(x=\frac{C}{A}\)
    2. At the y-intercept the x-coordinate is 0, (0, y).
      1. A(0) + By = C
      2. By = C
      3. \(y=\frac{C}{B}\)
  2. Plot the two points.
  3. Draw a line through the two points.

Example 4

Graph 3x − 4y = 12

Solution

Find the x- and y-intercepts.

x-intercept:

3x − 4(0) = 12

3x = 12

x = 4

The x-intercept is (4, 0).

y-intercept:

3(0) − 4y = 12

−4y = 12

y = −3

The y-intercept is (0, −3).

Plot those points and draw a line.

3x − 4y = 12
Figure 6: 3x − 4y = 12

Since horizontal lines have zero slope, their equation becomes y = b. Vertical lines are similar, but have the equation x = c.

Horizontal and Vertical Lines

Horizontal lines: y = b

Vertical lines: x = c

Example 5

Graph y = 3 and x = −2.

Solution

y = 3 is a horizontal line, so find a point where y = 3 and draw a horizontal line through it.

y = 3
Figure 7: y = 3

x = −2 is a vertical line, so find a point where x = −2 and draw a vertical line through it.

x = −2
Figure 8: x = −2

Practice Exercises

    Solve the equation for y.

  1. 2x + y = 4

    2. Solve the equation for y, then graph it by making a table.

  2. x − 3y = 7

    3. Graph the equation. If necessary, solve for y first.

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

    8. Find the x- and y-intercepts of the equation.

  8. 3x + 2y = 12

    9. Graph by finding the intercepts of the equation.

  9. 2x − 3y = 6
  10. x + y = 4
  11. 4x − 5y = 20
  12. y = −1
  13. x = 2

    14. Problem Solving

  14. The cost for a telephone call is 10 cents plus 5 cents per minute. This can be modeled by C = 5t + 10 where C is the cost of the call in cents and t is the time of the call in minutes. Graph the model.

    15. Mixed Review

  15. (0-04) Find the slope of the line through the points (−2, 4) and (3, −3).
  16. (0-04) Find the equation of the line passing through (−2, 4) and (3, −3).
  17. rain gauge
    Figure 9: Rain gauge (NASA/Henry Reges/ CoCoRaHS HQ)
    (0-04) During a long drenching rain, Sally decides to use a rain gauge to see how much rain is falling. At 1:00 PM her gauge reads 0.3 inches. At 3:00 PM it reads 1.8 inches. What is the rate that the rain is falling?
  18. (0-03) Solve |x + 5| = 8.
  19. (0-03) Solve 2|x − 2| ≥ 0.
  20. (0-01) Solve 2(x − 1) = 4x

Answers

  1. y = −2x + 4
  2. \(y=\frac{1}{3}x-\frac{7}{3}\),
  8. x = 4, y = 6
  15. \(m=-\frac{7}{5}\)
  16. \(y=-\frac{7}{5}x+\frac{6}{5}\)
  17. 0.75 in./hr.
  18. −13, 3
  19. All real numbers
  20. −1