### Graphing a Line in Slope Intercept Form

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

Let's graph the following equations.

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

First, the Cartesian plane is the \begin{align*}x-y\end{align*} 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*}intercept. By looking at the equation, \begin{align*}\frac{1}{3}\end{align*} is the slope and 4, or (0, 4), is the \begin{align*}y-\end{align*}intercept. To start graphing this line, plot the \begin{align*}y-\end{align*}intercept on the \begin{align*}y-\end{align*}axis.

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*}, so for \begin{align*}\frac{1}{3}\end{align*}, we will rise 1 and run 3 from the \begin{align*}y-\end{align*}intercept. 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*}

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

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*}intercept, and then use the slope to find a few more points.

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

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

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

So your bill for the month would be $93.75.

#### Example 2

Graph the following lines.

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

All answers are on the same grid below.

### Review

Graph the following lines in the Cartesian plane.

- \begin{align*}y = -2x -3\end{align*}
- \begin{align*}y = x + 4\end{align*}
- \begin{align*}y = \frac{1}{3}x - 1\end{align*}
- \begin{align*}y = 9\end{align*}
- \begin{align*}y = - \frac{2}{5}x + 7\end{align*}
- \begin{align*}y = \frac{2}{4}x - 5\end{align*}
- \begin{align*}y = -5x -2\end{align*}
- \begin{align*}y = -x\end{align*}
- \begin{align*}y = 4\end{align*}
- \begin{align*}x = -3\end{align*}
- \begin{align*}y = \frac{3}{2}x + 3\end{align*}
- \begin{align*}y = - \frac{1}{6}x - 8\end{align*}
- Graph \begin{align*}y = 4\end{align*} and \begin{align*}x = -6\end{align*} on the same set of axes. Where do they intersect?
- If you were to make a general rule for the lines \begin{align*}y = b\end{align*} and \begin{align*}x = a\end{align*}, where will they always intersect?
- The cost per month, \begin{align*}C\end{align*} (in dollars), of placing an ad on a website is \begin{align*}C = 0.25x + 50,\end{align*}where \begin{align*}x\end{align*} 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.