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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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Practice Graphs Using Slope-Intercept Form
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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.

### Guidance

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

#### Example A

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

Solution: First, the Cartesian plane is the xy\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 y\begin{align*}y-\end{align*}intercept. By looking at the equation, 13\begin{align*}\frac{1}{3}\end{align*} is the slope and 4, or (0, 4), is the y\begin{align*}y-\end{align*}intercept. To start graphing this line, plot the y\begin{align*}y-\end{align*}intercept on the y\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 riserun\begin{align*}\frac{rise}{run}\end{align*}, so for 13\begin{align*}\frac{1}{3}\end{align*}, we will rise 1 and run 3 from the y\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 y=13x+4\begin{align*}y = \frac{1}{3}x + 4\end{align*}.

#### Example B

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

Solution: 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 y\begin{align*}y-\end{align*}intercept, and then use the slope to find a few more points.

#### Example C

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

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

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

Intro Problem Revisit 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 y=7.5x+60\begin{align*}y=7.5x + 60\end{align*}. If you use 4.5 GB in a month, the total cost would be y=7.5(4.5)+60=93.75\begin{align*}y=7.5(4.5)+60=93.75\end{align*}.

So your bill for the month would be \$93.75.

### Guided Practice

Graph the following lines.

1. y=x+2\begin{align*}y = -x + 2\end{align*}

2. y=34x1\begin{align*}y = \frac{3}{4}x - 1\end{align*}

3. y=6\begin{align*}y = -6\end{align*}

All the answers are on the same grid below.

1. Plot (0, 2) and the slope is -1, which means you fall 1 and run 1.

2. Plot (0, -1) and then rise 3 and run 4 to the next point, (4, 2).

3. Plot -6 on the y\begin{align*}y-\end{align*}axis and draw a horizontal line.

### Explore More

Graph the following lines in the Cartesian plane.

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

### Vocabulary Language: English

Cartesian Plane

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

linear equation

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

Linear Function

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

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.