4.4 Understand slope of a line  (Page 2/14)

 Page 2 / 14

What is the slope of the line on the geoboard shown?

Solution

Use the definition of slope: $m=\frac{\text{rise}}{\text{run}}.$

Start at the left peg and count the spaces up and to the right to reach the second peg.

$\begin{array}{ccc}\text{The rise is 3.}\hfill & & \phantom{\rule{5em}{0ex}}m=\frac{3}{\text{run}}\hfill \\ \text{The run is 4.}\hfill & & \phantom{\rule{5em}{0ex}}m=\frac{3}{4}\hfill \\ & & \phantom{\rule{5em}{0ex}}\text{The slope is}\phantom{\rule{0.4em}{0ex}}\frac{3}{4}.\hfill \end{array}$

This means that the line rises 3 units for every 4 units of run.

What is the slope of the line on the geoboard shown?

$\frac{4}{3}$

What is the slope of the line on the geoboard shown?

$\frac{1}{4}$

What is the slope of the line on the geoboard shown?

Solution

Use the definition of slope: $m=\frac{\text{rise}}{\text{run}}.$

Start at the left peg and count the units down and to the right to reach the second peg.

$\begin{array}{cccc}\text{The rise is}\phantom{\rule{0.2em}{0ex}}-1.\hfill & & & \phantom{\rule{1.5em}{0ex}}m=\frac{-1}{\text{run}}\hfill \\ \text{The run is 3.}\hfill & & & \phantom{\rule{1.5em}{0ex}}m=\frac{-1}{3}\hfill \\ & & & \phantom{\rule{1.5em}{0ex}}m=-\frac{1}{3}\hfill \\ & & & \text{The slope is}\phantom{\rule{0.2em}{0ex}}-\frac{1}{3}.\hfill \end{array}$

This means that the line drops 1 unit for every 3 units of run.

What is the slope of the line on the geoboard?

$-\frac{2}{3}$

What is the slope of the line on the geoboard?

$-\frac{4}{3}$

Notice that in [link] the slope is positive and in [link] the slope is negative. Do you notice any difference in the two lines shown in [link] (a) and [link] (b)?

We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in [link] (a) is going up; it has positive slope    . The line in [link] (b) is going down; it has negative slope    .

Positive and negative slopes

Use a geoboard to model a line with slope $\frac{1}{2}$ .

Solution

To model a line on a geoboard, we need the rise and the run.

$\begin{array}{ccccc}\text{Use the slope formula.}\hfill & & \hfill \phantom{\rule{8em}{0ex}}m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}\frac{1}{2}.\hfill & & \hfill \phantom{\rule{8em}{0ex}}\frac{1}{2}& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \end{array}$

So, the rise is 1 and the run is 2.

Start at a peg in the lower left of the geoboard.

Stretch the rubber band up 1 unit, and then right 2 units.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is $\frac{1}{2}$ .

Model the slope $m=\frac{1}{3}$ . Draw a picture to show your results.

Model the slope $m=\frac{3}{2}$ . Draw a picture to show your results.

Use a geoboard to model a line with slope $\frac{-1}{4}.$

Solution

$\begin{array}{ccccc}\text{Use the slope formula.}\hfill & \phantom{\rule{9em}{0ex}}\hfill & \hfill m& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \\ \text{Replace}\phantom{\rule{0.2em}{0ex}}m\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}\frac{-1}{\phantom{\rule{0.4em}{0ex}}4}.\hfill & \phantom{\rule{9em}{0ex}}\hfill & \hfill \frac{-1}{\phantom{\rule{0.4em}{0ex}}4}& =\hfill & \frac{\text{rise}}{\text{run}}\hfill \end{array}$

So, the rise is $-1$ and the run is 4.

Since the rise is negative, we choose a starting peg on the upper left that will give us room to count down.

We stretch the rubber band down 1 unit, then go to the right 4 units, as shown.

The hypotenuse of the right triangle formed by the rubber band represents a line whose slope is $\frac{-1}{4}$ .

Model the slope $m=\frac{-2}{3}$ . Draw a picture to show your results.

Model the slope $m=\frac{-1}{3}$ . Draw a picture to show your results.

Use $m=\frac{\text{rise}}{\text{run}}$ To find the slope of a line from its graph

Now, we’ll look at some graphs on the $xy$ -coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.

To find the slope, we must count out the rise and the run. But where do we start?

We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

How to use $m=\frac{\text{rise}}{\text{run}}$ To find the slope of a line from its graph

Find the slope of the line shown.

Solution

Find the slope of the line shown.

$\frac{2}{5}$

Find the slope of the line shown.

$\frac{3}{4}$

Find the slope of a line from its graph using $m=\frac{\text{rise}}{\text{run}}.$

1. Locate two points on the line whose coordinates are integers.
2. Starting with the point on the left, sketch a right triangle, going from the first point to the second point.
3. Count the rise and the run on the legs of the triangle.
4. Take the ratio of rise to run to find the slope, $m=\frac{\text{rise}}{\text{run}}$ .

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