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# Slope in the Coordinate Plane

## Steepness of a line between two given points.

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Slope in the Coordinate Plane

What if you wanted to compare the steepness of two roofs? After completing this Concept, you'll be able to determine the steepness of lines in the coordinate plane using what you learned in Algebra I about slope.

### Guidance

Recall from Algebra I, the slope of the line between two points (x1, y1)\begin{align*}(x_1, \ y_1)\end{align*} and (x2, y2\begin{align*}(x_2, \ y_2\end{align*}) is m=(y2y1)(x2x1)\begin{align*}m=\frac{(y_2-y_1)}{(x_2-x_1)}\end{align*}.

Different Types of Slope:

#### Example A

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

Use the slope formula to determine the slope. Use (2, 2) as (x1, y1)\begin{align*}(x_1, \ y_1)\end{align*} and (4, 6) as (x2, y2)\begin{align*}(x_2, \ y_2)\end{align*}.

m=6242=42=2

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

#### Example B

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

m=232(8)=510=12\begin{align*}m=\frac{-2-3}{2-(-8)}= \frac{-5}{10}=-\frac{1}{2}\end{align*}

This is a negative slope.

#### Example C

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

m=1(1)3(5)=08=0

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

#### Example D

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

m=6233=40=undefined

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

Watch this video for help with the Examples above.

### Vocabulary

Slope is the steepness of a line. Two points (x1,y1)\begin{align*}(x_1, y_1)\end{align*} and (x2,y2)\begin{align*}(x_2, y_2)\end{align*} have a slope of m=(y2y1)(x2x1)\begin{align*}m = \frac{(y_2-y_1)}{(x_2-x_1)}\end{align*}.

### Guided Practice

Find the slope between the two given points:

1. (3, -4) and (3, 7)

2. (6, 1) and (4, 2)

3. (5, 7) and (11, 7)

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

2. The slope is (21)(46)=12\begin{align*}\frac{(2-1)}{(4-6)}=-\frac{1}{2}\end{align*}.

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

### Practice

Find the slope between the two given points.

1. (4, -1) and (-2, -3)
2. (-9, 5) and (-6, 2)
3. (7, 2) and (-7, -2)
4. (-6, 0) and (-1, -10)
5. (1, -2) and (3, 6)
6. (-4, 5) and (-4, -3)
7. (-2, 3) and (-2, -3)
8. (4, 1) and (7, 1)
9. (22, 37) and (-34, 56)
10. (13, 12) and (-23, 14)
11. (-4, 2) and (-16, 12)

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

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