11.4 Understand slope of a line  (Page 2/9)

Solution

Use the definition of slope.

$m=\frac{\text{rise}}{\text{run}}$

Start at the left peg and make a right triangle by stretching the rubber band up and to the right to reach the second peg.

Count the rise and the run as shown.

$\begin{array}{cccc}\text{The rise is}3\text{units}.\hfill & & & m=\frac{3}{\text{run}}\hfill \\ \text{The run is}4\text{units}.\hfill & & & m=\frac{3}{4}\hfill \\ & & & \text{The slope is}\frac{3}{4}.\hfill \end{array}$

Solution

Use the definition of slope.

$m=\frac{\text{rise}}{\text{run}}$

Start at the left peg and make a right triangle by stretching the rubber band to the peg on the right. This time we need to stretch the rubber band down to make the vertical leg, so the rise is negative.

$\begin{array}{cccc}\text{The rise is}\mathrm{-1}.\hfill & & & m=\frac{\mathrm{-1}}{\text{run}}\hfill \\ \text{The run is}3.\hfill & & & m=\frac{\mathrm{-1}}{3}\hfill \\ & & & m=-\frac{1}{3}\hfill \\ & & & \text{The slope is}-\frac{1}{3}.\hfill \end{array}$

Solution

To model a line with a specific slope on a geoboard, we need to know the rise and the run.

So, the rise is $1$ unit and the run is $2$ units.

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 with a slope of $\frac{1}{2}.$

Solution

So, the rise is $\mathrm{-1}$ 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 to the right $4$ units.

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

Solution

Locate two points on the graph, choosing points whose coordinates are integers. We will use $(0,\mathrm{-3})$ and $(5,1).$

Starting with the point on the left, $(0,\mathrm{-3}),$ sketch a right triangle, going from the first point to the second point, $(5,1).$

Count the rise on the vertical leg of the triangle.	The rise is 4 units.
Count the run on the horizontal leg.	The run is 5 units.
Use the slope formula.	$m=\frac{\text{rise}}{\text{run}}$
Substitute the values of the rise and run.	$m=\frac{4}{5}$
	The slope of the line is $\frac{4}{5}$ .

Notice that the slope is positive since the line slants upward from left to right.