11.4 Understand Slope of a Line

Learning objectives

Learning objectives

• Use geoboards to model slope

• Find the slope of a line from its graph

• Find the slope of horizontal and vertical lines

• Use the slope formula to find the slope of a line between two points

• Graph a line given a point and the slope Solve slope applications

As we’ve been graphing linear equations, we’ve seen that some lines slant up as they go from left to right and some lines slant down. Some lines are very steep and some lines are flatter. What determines whether a line slants up or down, and if its slant is steep or flat?

The steepness of the slant of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof and the grade of a highway or wheelchair ramp are just some examples in which you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.

Use Geoboards to Model Slope

In this section, we will explore the concepts of slope.

Using rubber bands on a geoboard gives a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

We’ll start by stretching a rubber band between two pegs to make a line as shown in Figure 11.17.

Does it look like a line?

Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle as shown in Figure 11.18. We carefully make a $90^{\circ}$ angle around the third peg, so that one side is vertical and the other is horizontal.

To find the slope of the line, we measure the distance along the vertical and horizontal legs of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in Figure 11.19.

To help remember the terms, it may help to think of the images shown in Figure 11.20.

On our geoboard, the rise is $2$ units because the rubber band goes up $2$ spaces on the vertical leg. See Figure 11.21.

What is the run? Be sure to count the spaces between the pegs rather than the pegs themselves! The rubber band goes across $3$ spaces on the horizontal leg, so the run is $3$ units.

The slope of a line is the ratio of the rise to the run. So the slope of our line is $\frac{2}{3}.$ In mathematics, the slope is always represented by the letter $m.$

What is the slope of the line on the geoboard in Figure 11.21?

When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. Then we stretch the rubber band to form a right triangle.

If we start by going up the rise is positive, and if we stretch it down the rise is negative. We will count the run from left to right, just like you read this paragraph, so the run will be positive.

Since the slope formula has rise over run, it may be easier to always count out the rise first and then the run.

Example 11.30

Solution

1. Use the slope definition. Slope is the ratio $m=\frac{\mathrm{rise}}{\mathrm{run}}$, measured along the vertical and horizontal legs of a right triangle.
2. Count the vertical leg. From the lower peg to the higher peg, the rubber band rises $3$ units, so the rise is $3$.
3. Count the horizontal leg. From the left peg to the right peg, the rubber band travels $4$ units, so the run is $4$.
4. Divide rise by run. Substituting the counted values gives $m=\frac{3}{4}$, and the positive ratio agrees with the line rising to the right.

Takeaway: Slope is rise over run, so counting $3$ units up and $4$ units right between two pegs gives $m=\frac{3}{4}$.

Example 11.31

Solution

1. Use the slope definition. The slope is $m=\frac{\mathrm{rise}}{\mathrm{run}}$, with the direction of travel determining the signs.
2. Measure the vertical change. Starting at the left peg, the band goes down $1$ unit, so the rise is $-1$.
3. Measure the horizontal change. The band travels $3$ units to the right, so the run is $3$.
4. Divide rise by run. The ratio is $m=\frac{-1}{3}=-\frac{1}{3}$, and the negative sign agrees with the line falling from left to right.

Takeaway: When the vertical leg of the triangle goes down, the rise is negative: down $1$ and right $3$ gives $m=-\frac{1}{3}$.

Notice that in the first example, the slope is positive and in the second example the slope is negative. Do you notice any difference in the two lines shown in Figure 11.22.

As you read from left to right, the line in Figure A, is going up; it has positive slope. The line Figure B is going down; it has negative slope.

Example 11.32

Solution

1. Interpret the slope fraction. The slope $\frac{1}{2}$ means a rise of $1$ unit for every run of $2$ units because the numerator gives rise and the denominator gives run.
2. Choose a starting peg. Begin at a peg near the lower-left corner so the band can move upward and to the right without leaving the board.
3. Stretch the band. Move upward $1$ peg and then right $2$ pegs, keeping the band's vertical and horizontal changes equal to the model.
4. Check the ratio. The rubber band's hypotenuse has rise $1$ and run $2$, so $\frac{1}{2}=\frac{\mathrm{rise}}{\mathrm{run}}$ confirms the modeled slope.

Takeaway: To model a line with slope $\frac{1}{2}$, move up $1$ unit and right $2$ units from any starting peg — the numerator is the rise and the denominator the run.

Example 11.33

Solution

1. Interpret the negative slope. The value $-\frac{1}{4}$ means the line falls $1$ unit for every $4$ units it travels to the right.
2. Choose a starting peg. Begin at a peg near the upper-left corner so the band has room to move downward and rightward.
3. Stretch the band. Move down $1$ peg and then right $4$ pegs, keeping the vertical change negative and the horizontal change positive.
4. Check the ratio. The hypotenuse has rise $-1$ and run $4$, so $\frac{-1}{4}=-\frac{1}{4}$ confirms that it models the required slope.

Takeaway: A slope of $-\frac{1}{4}$ means down $1$ and right $4$; the sign lives in the rise, so start high enough on the geoboard to count downward.

Find the Slope of a Line from its Graph

Now we’ll look at some graphs on a coordinate grid 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 any two points on the line. We try to choose points with coordinates that are integers to make our calculations easier. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.

Example 11.34

Solution

1. Choose two integer points. Use $(0,-3)$ and $(5,1)$, whose coordinates can be read exactly from the graph.
2. Draw a right triangle. From the left point $(0,-3)$, move horizontally $5$ units to $x=5$ and vertically $4$ units up to the point $(5,1)$.
3. Measure the legs. The horizontal leg is the run $5$, and the vertical leg is the rise $4$ because the line moves upward from left to right.
4. Calculate the slope. Dividing rise by run gives $m=\frac{4}{5}$, which is positive because the line rises from left to right.

Takeaway: Using the integer points $(0,-3)$ and $(5,1)$, the rise is $4$ and the run is $5$, so the line has positive slope $\frac{4}{5}$.

Example 11.35

Solution

1. Choose the first pair of points. Use $(0,5)$ and $(3,3)$, both of which have integer coordinates on the line.
2. Measure the first triangle. Moving right $3$ and down $2$ gives run $3$ and rise $-2$, so $m=\frac{-2}{3}$.
3. Choose a second pair of points. Use $(-3,7)$ and $(6,1)$ to verify that the slope does not depend on the selected points.
4. Measure the second triangle. Moving right $9$ and down $6$ gives run $9$ and rise $-6$, so $m=\frac{-6}{9}$.
5. Compare the ratios. Since $\frac{-6}{9}=-\frac{2}{3}$, both pairs give the same negative slope, and the line falls from left to right.

Takeaway: Slope belongs to the line, not the points chosen: $(0,5)$ and $(3,3)$ give $\frac{-2}{3}$, while $(-3,7)$ and $(6,1)$ give $\frac{-6}{9}$ — both equal $-\frac{2}{3}$.

The lines in the previous examples had $y$-intercepts with integer values, so it was convenient to use the y-intercept as one of the points we used to find the slope. In the next example, the $y$-intercept is a fraction. The calculations are easier if we use two points with integer coordinates.

Example 11.36

Solution

1. Choose two integer points. Use $(2,3)$ and $(7,6)$, whose coordinates can be read exactly from the graph.
2. Move from left to right. Begin at $(2,3)$ and travel horizontally $5$ units to $x=7$, so the run is $5$.
3. Measure the vertical change. The line rises from $y=3$ to $y=6$, so the rise is $3$.
4. Calculate the slope. Dividing rise by run gives $m=\frac{3}{5}$, which is positive because the line rises from left to right.

Takeaway: From $(2,3)$ to $(7,6)$ the rise is $3$ and the run is $5$, so the slope of the line is $\frac{3}{5}$.

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

• horizontal line $y=b;$ all the $y$-coordinates are the same.

• vertical line $x=a;$ all the $x$-coordinates are the same.

So how do we find the slope of the horizontal line $y=4?$ One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens in Figure 11.24. We’ll use the two points $(0,\,4)$ and $(3,\,4)$ to count the rise and run.

The slope of the horizontal line $y=4$ is $0.$

All horizontal lines have slope $0$. When the $y$-coordinates are the same, the rise is $0$.

Now we’ll consider a vertical line, such as the line $x=3$, shown in Figure 11.25. We’ll use the two points $(3,\,0)$ and $(3,\,2)$ to count the rise and run.

But we can’t divide by $0.$ Division by $0$ is undefined. So we say that the slope of the vertical line $x=3$ is undefined. The slope of all vertical lines is undefined, because the run is $0.$

Example 11.37

Solution

1. Analyze the vertical line. The line $x=8$ contains points such as $(8,0)$ and $(8,1)$; moving between them has rise $1$ and run $0$, so rise over run would require division by $0$ and the slope is undefined.
2. Analyze the horizontal line. The line $y=-5$ contains points such as $(0,-5)$ and $(1,-5)$; moving from left to right has rise $0$ and run $1$, so the slope is $\frac{0}{1}=0$.
3. State both results. Therefore $x=8$ has undefined slope, while $y=-5$ has slope $0$.

Takeaway: A vertical line has zero run and undefined slope, whereas a horizontal line has zero rise and slope $0$.

Use the Slope Formula to find the Slope of a Line between Two Points

Sometimes we need to find the slope of a line between two points and we might not have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but there is a way to find the slope without graphing.

Before we get to it, we need to introduce some new algebraic notation. We have seen that an ordered pair $(x,y)$ gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol $(x,y)$ be used to represent two different points?

Mathematicians use subscripts to distinguish between the points. A subscript is a small number written to the right of, and a little lower than, a variable.

• $({x}_{1},{y}_{1})\,\mathrm{read}\,x\,\mathrm{sub}\,1,y\,\mathrm{sub}\,1$

• $({x}_{2},{y}_{2})\,\mathrm{read}\,x\,\mathrm{sub}\,2,y\,\mathrm{sub}\,2$

We will use $({x}_{1},{y}_{1})$ to identify the first point and $({x}_{2},{y}_{2})$ to identify the second point. If we had more than two points, we could use $({x}_{3},{y}_{3}),({x}_{4},{y}_{4}),$ and so on.

To see how the rise and run relate to the coordinates of the two points, let’s take another look at the slope of the line between the points $(2,3)$ and $(7,6)$ in Figure 11.26.

Since we have two points, we will use subscript notation.

On the graph, we counted the rise of $3.$ The rise can also be found by subtracting the $y\mathrm{-coordinates}$ of the points.

We counted a run of $5.$ The run can also be found by subtracting the $x\mathrm{-coordinates}.$

We’ve shown that $m=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$ is really another version of $m=\frac{\mathrm{rise}}{\mathrm{run}}.$ We can use this formula to find the slope of a line when we have two points on the line.

Say the formula to yourself to help you remember it:

Example 11.38

Solution

1. Label the points consistently. Let point 1 be $(1,2)$ and point 2 be $(4,5)$, so $(x_1,y_1)=(1,2)$ and $(x_2,y_2)=(4,5)$.
2. Substitute into the slope formula. Using $m=\frac{y_2-y_1}{x_2-x_1}$ gives $m=\frac{5-2}{4-1}$.
3. Simplify the differences. The numerator is $3$ and the denominator is $3$, so $m=\frac{3}{3}=1$.
4. Check with rise over run. Moving from $(1,2)$ to $(4,5)$ rises $3$ units and runs $3$ units, which also gives slope $1$.

Takeaway: The slope formula applied to $(1,2)$ and $(4,5)$ gives $m=\frac{5-2}{4-1}=\frac{3}{3}=1$, matching the rise of $3$ over a run of $3$ counted on the graph.

How do we know which point to call $1 and which to call $2? Let’s find the slope again, this time switching the names of the points to see what happens. Since we will now be counting the run from right to left, it will be negative.

The slope is the same no matter which order we use the points.

Example 11.39

Solution

1. Label the two points in order. Calling $(-2,-3)$ point $1$ and $(-7,4)$ point $2$ gives $x_1=-2$, $y_1=-3$, $x_2=-7$, and $y_2=4$, and the same order must be used in both differences.
2. Write the slope formula. The slope is $m=\frac{y_2-y_1}{x_2-x_1}$, the signed change in $y$ over the signed change in $x$.
3. Subtract the $y$-values. Substituting the $y$-coordinates gives $m=\frac{4-(-3)}{x_2-x_1}=\frac{7}{x_2-x_1}$.
4. Subtract the $x$-values. Substituting the $x$-coordinates gives $m=\frac{7}{-7-(-2)}=\frac{7}{-5}$.
5. Simplify the sign. A negative denominator moves the sign to the front, so $m=-\frac{7}{5}$, which matches the rise of $-7$ over a run of $5$ on the graph.

Takeaway: With $(-2,-3)$ first and $(-7,4)$ second, $m=\frac{4-(-3)}{-7-(-2)}=\frac{7}{-5}=-\frac{7}{5}$ — subtracting in the same order above and below keeps the sign correct.

Graph a Line Given a Point and the Slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Another method we can use to graph lines is the point-slope method. Sometimes, we will be given one point and the slope of the line, instead of its equation. When this happens, we use the definition of slope to draw the graph of the line.

Example 11.40

Solution

1. Plot the given point. Mark $(1,-1)$ on the coordinate grid as the starting point of the line.
2. Read the rise and run. The slope $m=\frac{3}{4}$ means $\frac{\mathrm{rise}}{\mathrm{run}}=\frac{3}{4}$, so the rise is $3$ and the run is $4$.
3. Mark a second point. From $(1,-1)$ count $3$ units up and $4$ units right, which lands on $(5,2)$.
4. Draw the line. Connect $(1,-1)$ and $(5,2)$ with a straight line and add arrows at both ends to show that the line continues.
5. Check the slope. Using the two plotted points, $m=\frac{2-(-1)}{5-1}=\frac{3}{4}$, so the graph matches the given slope.

Takeaway: From the point $(1,-1)$, the slope $\frac{3}{4}$ directs the next point up $3$ and right $4$, and two points determine the line.

Example 11.41

Solution

1. Plot the $y$-intercept. Mark $(0,2)$, the point where the line crosses the $y$-axis.
2. Write the slope as rise over run. The slope $m=-\frac{2}{3}$ means a rise of $-2$ and a run of $3$, so the line moves down $2$ units for every $3$ units to the right.
3. Mark a second point. From $(0,2)$ count $2$ units down and $3$ units right, which lands on $(3,0)$, the $x$-intercept.
4. Draw the line. Connect $(0,2)$ and $(3,0)$ with a straight line and extend it with arrows at both ends.
5. Check the slope. The two points give $m=\frac{0-2}{3-0}=-\frac{2}{3}$, confirming the line has the required slope.

Takeaway: Starting at the $y$-intercept $(0,2)$, the slope $-\frac{2}{3}$ moves down $2$ and right $3$ to the next point of the line.

Example 11.42

Solution

1. Plot the given point. Mark $(-1,-3)$ on the coordinate grid as a point on the line.
2. Write the integer slope as a fraction. The slope $m=4$ can be written $m=\frac{4}{1}$, so the rise is $4$ and the run is $1$.
3. Mark a second point. From $(-1,-3)$ count $4$ units up and $1$ unit right, which lands on $(0,1)$.
4. Draw the line. Connect $(-1,-3)$ and $(0,1)$ with a straight line and add arrows at both ends.
5. Check the slope. From the plotted points, $m=\frac{1-(-3)}{0-(-1)}=\frac{4}{1}=4$, so the line matches the given slope.

Takeaway: An integer slope is a fraction over $1$: $m=4=\frac{4}{1}$ means from $(-1,-3)$ go up $4$ and right $1$.

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

Example 11.43

Solution

1. Read the rise and run from the figure. The roof rises $9$ feet over a horizontal run of $18$ feet, so the rise is $9$ ft and the run is $18$ ft.
2. Write the slope formula. The pitch of a roof is its slope, so use $m=\frac{\mathrm{rise}}{\mathrm{run}}$.
3. Substitute the measurements. Both measurements are in feet, so no conversion is needed: $m=\frac{9}{18}$.
4. Simplify the fraction. Dividing the numerator and denominator by $9$ gives $m=\frac{1}{2}$, so the roof rises one foot for every two feet of run.

Takeaway: Roof pitch is just slope, so a rise of $9$ ft over a run of $18$ ft gives $m=\frac{9}{18}=\frac{1}{2}$ — steep enough for snow to slide off.

Have you ever thought about the sewage pipes going from your house to the street? Their slope is an important factor in how they take waste away from your house.

Example 11.44

Solution

1. Identify the rise and the run. The pipe drops $\frac{1}{4}$ inch over a run of $1$ foot, so the rise is $-\frac{1}{4}$ inch and the run is $1$ foot.
2. Convert the units. Rise and run must use the same unit, and $1$ foot is $12$ inches, so the run is $12$ inches.
3. Write the slope formula. Substituting the same-unit measurements gives $m=\frac{-\frac{1}{4}\,\mathrm{in.}}{12\,\mathrm{in.}}$.
4. Simplify the fraction. Dividing $\frac{1}{4}$ by $12$ multiplies by $\frac{1}{12}$, so $m=-\frac{1}{48}$; the negative sign records that the pipe descends as it runs.

Takeaway: The pipe drops $\frac{1}{4}$ in. over a run of $1$ ft $=12$ in., so its slope is $\frac{-\frac{1}{4}}{12}=-\frac{1}{48}$; rise and run must be in the same units before dividing.

Section 11.4 Exercises

Section 11.4 Exercises

Use Geoboards to Model Slope

In the following exercises, find the slope modeled on each geoboard.

In the following exercises, model each slope. Draw a picture to show your results.

Find the Slope of a Line from its Graph

In the following exercises, find the slope of each line shown.

Find the Slope of Horizontal and Vertical Lines

In the following exercises, find the slope of each line.

Use the Slope Formula to find the Slope of a Line between Two Points

In the following exercises, use the slope formula to find the slope of the line between each pair of points.

Graph a Line Given a Point and the Slope

In the following exercises, graph the line given a point and the slope.

Solve Slope Applications

In the following exercises, solve these slope applications.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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

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

1. Find the slope of the line:

2. Find the slope of the line:

1. What is the slope of the line on the geoboard?

2. What is the slope of the line on the geoboard?

1. Find the slope of the line:

2. Find the slope of the line:

1. Find the slope of the line:

2. Find the slope of the line:

1. Find the slope of the line: $x=-4.$

2. Find the slope of the line through the pair of points: $(-3,4)$ and $(2,-1).$

1. Find the slope given rise and run: A roof with a rise $=14$ and run $=24.$

2. Find the slope given rise and run: A roof with a rise $=15$ and run $=36.$

1. Find the slope of the pipe: The pipe slopes down $\frac{1}{3}$ inch per foot.

2. Find the slope of the pipe: The pipe slopes down $\frac{3}{4}$ inch per yard.

1. $x=4$

2. $x=-5$

1. $(1,4),(3,9)$

2. $(0,3),(4,6)$

1. $(2,5),(4,0)$

2. $(-3,3),(2,-5)$

1. $(-1,-2),(2,5)$

2. $(4,-5),(1,-2)$

1. Wheelchair ramp The rules for wheelchair ramps require a maximum $1$ inch rise for a $12$ inch run.

1. ⓐ What run must the ramp have to accommodate a $\mathrm{24-inch}$ rise to the door?

2. ⓑ Draw a model of this ramp.

2. What does the sign of the slope tell you about a line?

1. Slope of a roof A fairly easy way to determine the slope is to take a $\mathrm{12-inch}$ level and set it on one end on the roof surface. Then take a tape measure or ruler, and measure from the other end of the level down to the roof surface. You can use these measurements to calculate the slope of the roof. What is the slope of the roof in this picture?

2. Road grade A local road has a grade of $\mathrm{6\%}.$ The grade of a road is its slope expressed as a percent.

1. ⓐ Find the slope of the road as a fraction and then simplify the fraction.

2. ⓑ What rise and run would reflect this slope or grade?

1. Why is the slope of a vertical line undefined?