### Slope in the Coordinate Plane

Recall from Algebra I, the slope of the line between two points \begin{align*}(x_1, \ y_1)\end{align*} and \begin{align*}(x_2, \ y_2\end{align*}) is \begin{align*}m=\frac{(y_2-y_1)}{(x_2-x_1)}\end{align*}.

Different Types of Slope:

#### Calculating Slope given a Graph

What is the slope of the line through (2, 2) and (4, 6)?

#### Calculating the Slope given Points

Use the slope formula to determine the slope. Use (2, 2) as \begin{align*}(x_1, \ y_1)\end{align*} and (4, 6) as \begin{align*}(x_2, \ y_2)\end{align*}.

\begin{align*}m=\frac{6-2}{4-2}=\frac{4}{2}=2\end{align*}

Therefore, the slope of this line is 2. This slope is positive.

Find the slope between (-8, 3) and (2, -2).

\begin{align*}m=\frac{-2-3}{2-(-8)}= \frac{-5}{10}=-\frac{1}{2}\end{align*}

This is a negative slope.

#### Calculating the Slope of a Horizontal Line

Find the slope between (-5, -1) and (3, -1).

\begin{align*}m=\frac{-1-(-1)}{3-(-5)}= \frac{0}{8}=0\end{align*}

Therefore, the slope of this line is 0, which means that it is a horizontal line. Horizontal lines always pass through the \begin{align*}y-\end{align*}axis. Notice that the \begin{align*}y-\end{align*}coordinate for both points is -1. In fact, the \begin{align*}y-\end{align*}coordinate for *any* point on this line is -1. This means that the horizontal line must cross \begin{align*}y = -1\end{align*}.

#### Calculating an Undefined Slope

What is the slope of the line through (3, 2) and (3, 6)?

\begin{align*}m=\frac{6-2}{3-3}=\frac{4}{0}=undefined\end{align*}

Therefore, the slope of this line is undefined, which means that it is a *vertical* line. Vertical lines always pass through the \begin{align*}x-\end{align*}axis. Notice that the \begin{align*}x-\end{align*}coordinate for both points is 3. In fact, the \begin{align*}x-\end{align*}coordinate for *any* point on this line is 3. This means that the vertical line must cross \begin{align*}x = 3\end{align*}.

### Examples

Find the slope between the two given points:

#### Example 1

(3, -4) and (3, 7)

These two points create a vertical line, so the slope is undefined.

#### Example 2

(6, 1) and (4, 2)

The slope is \begin{align*}\frac{(2-1)}{(4-6)}=-\frac{1}{2}\end{align*}.

#### Example 3

(5, 7) and (11, 7)

These two points create a horizontal line, so the slope is zero.

### Interactive Practice

### Review

Find the slope between the two given points.

- (4, -1) and (-2, -3)
- (-9, 5) and (-6, 2)
- (7, 2) and (-7, -2)
- (-6, 0) and (-1, -10)
- (1, -2) and (3, 6)
- (-4, 5) and (-4, -3)
- (-2, 3) and (-2, -3)
- (4, 1) and (7, 1)
- (22, 37) and (-34, 56)
- (13, 12) and (-23, 14)
- (-4, 2) and (-16, 12)

For 12-15, determine if the statement is true or false.

- If you know the slope of a line you will know whether it is pointing up or down from left to right.
- Vertical lines have a slope of zero.
- Horizontal lines have a slope of zero.
- The larger the absolute value of the slope, the less steep the line.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 3.7.