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Graphs Using Slope-Intercept Form

Use the y-intercept and the 'rise over run' to graph a line

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Graph a Line in Slope-Intercept Form
The cost per month for a cell-phone plan is $60 plus $7.50 for every gigabyte (GB) of data you use. (For billing purposes, actual usage is rounded to the nearest one-quarter GB.) Write an equation for the cost of the data plan and determine how much your bill will be if you use 4.5 GB of data in a month.

Graphing a Line in Slope Intercept Form

Recall that the equation of a line is \begin{align*}y = mx + b\end{align*}y=mx+b, where \begin{align*}m\end{align*}m is the slope and \begin{align*}b\end{align*}b is the \begin{align*}y-\end{align*}yintercept. From these two pieces of information we can graph any line.

Let's graph the following equations.

  1. Graph \begin{align*}y = \frac{1}{3}x + 4\end{align*}y=13x+4 on the Cartesian plane.

First, the Cartesian plane is the \begin{align*}x-y\end{align*}xy plane. Typically, when graphing lines, draw each axis from -10 to 10. To graph this line, you need to find the slope and \begin{align*}y-\end{align*}yintercept. By looking at the equation, \begin{align*}\frac{1}{3}\end{align*}13 is the slope and 4, or (0, 4), is the \begin{align*}y-\end{align*}yintercept. To start graphing this line, plot the \begin{align*}y-\end{align*}yintercept on the \begin{align*}y-\end{align*}yaxis.

Now we need to use the slope to find the next point on the line. Recall that the slope is also \begin{align*}\frac{rise}{run}\end{align*}riserun, so for \begin{align*}\frac{1}{3}\end{align*}13, we will rise 1 and run 3 from the \begin{align*}y-\end{align*}yintercept. Do this a couple of times to get at least three points.

Now that we have three points, connect them to form the line \begin{align*}y = \frac{1}{3}x + 4.\end{align*}y=13x+4.

  1. Graph \begin{align*}y = -4x -5.\end{align*}y=4x5.

Now that the slope is negative, the vertical distance will “fall” instead of rise. Also, because the slope is a whole number, we need to put it over 1. Therefore, for a slope of -4, the line will fall 4 and run 1 OR rise 4 and run backward 1. Start at the \begin{align*}y-\end{align*}yintercept, and then use the slope to find a few more points.

  1. Graph \begin{align*}x = 5\end{align*}x=5.

Any line in the form \begin{align*}x = a\end{align*}x=a is a vertical line. To graph any vertical line, plot the value, in this case 5, on the \begin{align*}x-\end{align*}xaxis. Then draw the vertical line.

To graph a horizontal line, \begin{align*}y = b\end{align*}y=b, it will be the same process, but plot the value given on the \begin{align*}y-\end{align*}yaxis and draw a horizontal line.

Examples

Example 1

Earlier, you were asked to write an equation for the cost of a data plan, and determine how much your bill will be if you use 4.5 GB of data a month. 

If x is the number of GB of data you use in a month and y is the total cost you pay, then the equation for the cell-phone plan would be \begin{align*}y=7.5x + 60\end{align*}y=7.5x+60. If you use 4.5 GB in a month, the total cost would be \begin{align*}y=7.5(4.5)+60=93.75\end{align*}y=7.5(4.5)+60=93.75.

So your bill for the month would be $93.75.

Example 2

Graph the following lines.

  1. \begin{align*}y = -x + 2\end{align*}y=x+2 (Plot (0, 2) and the slope is -1, which means you fall 1 and run 1)
  2. \begin{align*}y = \frac{3}{4}x - 1\end{align*}y=34x1 (Plot (0, -1) and then rise 3 and run 4 to the next point, (4, 2))
  3. \begin{align*}y = -6\end{align*}y=6 (Plot -6 on the \begin{align*}y-\end{align*}yaxis and draw a horizontal line)

All answers are on the same grid below.

Review

Graph the following lines in the Cartesian plane.

  1. \begin{align*}y = -2x -3\end{align*}y=2x3
  2. \begin{align*}y = x + 4\end{align*}y=x+4
  3. \begin{align*}y = \frac{1}{3}x - 1\end{align*}y=13x1
  4. \begin{align*}y = 9\end{align*}y=9
  5. \begin{align*}y = - \frac{2}{5}x + 7\end{align*}y=25x+7
  6. \begin{align*}y = \frac{2}{4}x - 5\end{align*}y=24x5
  7. \begin{align*}y = -5x -2\end{align*}y=5x2
  8. \begin{align*}y = -x\end{align*}y=x
  9. \begin{align*}y = 4\end{align*}y=4
  10. \begin{align*}x = -3\end{align*}x=3
  11. \begin{align*}y = \frac{3}{2}x + 3\end{align*}y=32x+3
  12. \begin{align*}y = - \frac{1}{6}x - 8\end{align*}y=16x8
  13. Graph \begin{align*}y = 4\end{align*}y=4 and \begin{align*}x = -6\end{align*}x=6 on the same set of axes. Where do they intersect?
  14. If you were to make a general rule for the lines \begin{align*}y = b\end{align*}y=b and \begin{align*}x = a\end{align*}x=a, where will they always intersect?
  15. The cost per month, \begin{align*}C\end{align*}C (in dollars), of placing an ad on a website is \begin{align*}C = 0.25x + 50,\end{align*}C=0.25x+50,where \begin{align*}x\end{align*}x is the number of times someone clicks on your link. How much would it cost you if 500 people clicked on your link?

Answers for Review Problems

To see the Review answers, open this PDF file and look for section 2.6. 

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Vocabulary

Cartesian Plane

The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.

linear equation

A linear equation is an equation between two variables that produces a straight line when graphed.

Linear Function

A linear function is a relation between two variables that produces a straight line when graphed.

Slope-Intercept Form

The slope-intercept form of a line is y = mx + b, where m is the slope and b is the y-intercept.

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