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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

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.

  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.

Review (Answers)

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

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