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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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Graphs Using Slope-Intercept Form

Suppose a company had the slope and \begin{align*}y\end{align*}-intercept of a line representing revenue based on units sold. How could the company graph this line? What if another company graphed its own revenue line, and it was parallel to the first company's line? What would this say about the per-unit revenue of the two companies?

### Graphs Using Slope-Intercept Form

The slope of a line describes how to navigate from one point on the line to another, and the \begin{align*}y\text{-intercept}\end{align*} of a line describes the point where the line crosses the \begin{align*}y\text{-axis}.\end{align*}Once you have these two bits of information, graphing a line is easy.

#### Let's graph the following equations:

1.   \begin{align*}y=2x+5.\end{align*}

The equation is in slope-intercept form. To graph the solutions to this equation, you should start at the \begin{align*}y\text{-intercept}.\end{align*} Then, using the slope, find a second coordinate. Finally, draw a line through the ordered pairs.

1.  \begin{align*}y=-3x+5.\end{align*}

Using the definition of slope-intercept form, this equation has a \begin{align*}y\text{-intercept}\end{align*} of (0, 5) and a slope of \begin{align*}\frac{\text{-}3}{1}\end{align*}.

#### Slopes of Parallel Lines

Parallel lines will never intersect, or cross. The only way for two lines never to cross is if the method of finding additional coordinates is the same.

Therefore, it's true that parallel lines have the same slope.

#### Let's use this information to determine the slope of any line parallel to \begin{align*}y=\text{-}3x+5\end{align*}:

Because parallel lines have the same slope, the slope of any line parallel to \begin{align*}y=\text{-}3x+5\end{align*} must also be -3.

### Examples

#### Example 1

Earlier, you were told that a company had the slope and \begin{align*}y\end{align*}-intercept of a line representing revenue based on units sold. How could the company graph this line? What if another company graphed its own revenue line and noticed that it was parallel to the first company's line? What would this say about the per-unit revenue of the two companies?

The company could graph the line by graphing the \begin{align*}y\end{align*}-intercept first and then using the slope to find a second point on the graph. If another company found that its line was parallel to the first company's line, then that would mean that the per-unit revenue of the two companies is the same. The only difference may be in the starting revenue of the units (the \begin{align*}y\end{align*}-intercept).

#### Example 2

Graph \begin{align*}y=-\frac{2}{5}x\end{align*} by graphing the \begin{align*}y\end{align*}-intercept first, and then using the slope to find a second point to graph.

First, graph the \begin{align*}y\end{align*}-intercept, which is \begin{align*}(0,0)\end{align*}.

Next, the slope is \begin{align*}-\frac{2}{5}\end{align*}. The negative can go along with the denominator or numerator. Either way, you will get a slope on the same line. Let the rise be 2 and the run be -5. This means, from the starting point, go to the left 5, and then go up 2:

### Review

Plot the following equations on a graph.

1. \begin{align*}y=2x+5\end{align*}
2. \begin{align*}y=-0.2x+7\end{align*}
3. \begin{align*}y=-x\end{align*}
4. \begin{align*}y=3.75\end{align*}
5. \begin{align*}\frac{2}{7} x-4=y\end{align*}
6. \begin{align*}y=-4x+13\end{align*}
7. \begin{align*}-2+\frac{3}{8} x=y\end{align*}
8. \begin{align*}y=\frac{1}{2}+2x\end{align*}

In 9 – 16, state the slope of a line parallel to the given line.

1. \begin{align*}y=2x+5\end{align*}
2. \begin{align*}y=-0.2x+7\end{align*}
3. \begin{align*}y=-x\end{align*}
4. \begin{align*}y=3.75\end{align*}
5. \begin{align*}y=-\frac{1}{5}x-11\end{align*}
6. \begin{align*}y=-5x+5\end{align*}
7. \begin{align*}y=-3x+11\end{align*}
8. \begin{align*}y=3x+3.5\end{align*}

To see the Review answers, open this PDF file and look for section 4.10.

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### Vocabulary Language: English

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.