3.4 Slope and Rate of Change

Scott McDaniel

3.4 Slope and Rate of Change

Learning Objectives

After completing this section, you should be able to:

Use $m=\frac{\text{rise}}{\text{run}}$ to 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 between two points.
Solve real-world problems involving slope (rate of change).
Find the slope of a line from an equation.
Use slopes to identify parallel and perpendicular lines.

Introduction

The slope of a line describes how steep the line is and which direction it tilts.

It measures rate of change — how much $y$ changes for a given change in $x$ .

The formula for slope is:

$m=\frac{\text{rise}}{\text{run}}=\frac{\text{change in }y}{\text{change in }x}$

Slope compares the vertical change (rise) to the horizontal change (run) between two points on a line.

Finding Slope from a Graph

To find slope from a graph:

Locate two clear points on the line.
Count the vertical change (rise).
Count the horizontal change (run).
Divide: $\text{slope}=\frac{\text{rise}}{\text{run}}$ .

Example 3.4.1a— Slope from a Graph

Find the slope of the line passing through the points $(1,3)$ and $(2,5)$ .

Line through two points (1,3) and (2,5) — **Figure 3.4A**

Solution

Rise = $5-3=2$

Run = $2-1=1$

$m=\frac{2}{1}=2$

The slope is 2, meaning the line rises 2 units for every 1 unit to the right.

Line through the points (1,3) and (2,5) showing rise 2 and run 1. — **Figure 3.4B**

Example 3.4.1b — Slope from a Graph

Find the slope of the line passing through the points
$(0,4)$ and $(3,2)$ .

Graph of a line passing through the points (0,4) and (3,2). — **Figure 3.4C**

Solution

Start at the point $(0,4)$ on the graph (you may start at either point).
Move along the grid to reach the point $(3,2)$ .

From $(0,4)$ , move down 2 units and right 3 units.

Since the line goes down as you move to the right, the slope is negative.
The slope is $-\frac{2}{3}$ .

Graph showing a vertical drop of 2 units and a horizontal run of 3 units between the points. — **Figure 3.4D**

Try It

Example 3.4.2 — Horizontal and Vertical Slopes

Determine the slope of each line.

a) y = 4

b) x = -2

Solution

a) For $y=4$ , the slope is $0$ (horizontal line).

Linear graph y = 4. A line through the y-intercept of 4 — **Figure 3.4E**

b) For $x=-2$ , the slope is undefined (vertical line).

Graph of vertical line x = -2 — **Figure 3.4F**

Summary

**Table 3.4A**
Line Type	Description	Slope
Rising line	Increases from left to right	Positive
Falling line	Decreases from left to right	Negative
Horizontal line	Flat (no rise)	$0$
Vertical line	No run	Undefined

Finding Slope Between Two Points

You won’t always have a nice graph with clear points. Often you are just given 2 points.

If you know two points $(x_1,y_1)$ and $(x_2,y_2)$ , use the slope formula:

$m=\frac{y_2-y_1}{x_2-x_1}$

Example 3.4.3 — Slope Formula

Find the slope between $(2,5)$ and $(6,9)$ .

Solution

$m=\frac{9-5}{6-2}=\frac{4}{4}=1$

Line through the points (2,5) and (6,9) — **Figure 3.4G**

Example 3.4.4

Find the slope between $(1,7)$ and $(4,-2)$ .

Solution

$m=\frac{-2-7}{4-1}=\frac{-9}{3}=-3$

The line falls 3 units for every 1 unit to the right.

Line through (1, 7) and (4, -2). — **Figure 3.4H**

Example 3.4.5 — Fractional Slope

Find the slope between $(0,-2)$ and $(4,5)$ .

Solution

$m=\frac{5-(-2)}{4-0}=\frac{7}{4}$

Line through (0,-2) and (4,5) showing slope up 7 right 4 for a slope of 7/4 — **Figure 3.4I**

Exercises

Slope in Real-World Contexts

Slope often represents a real-world scenarios

Roof Pitch

A house’s roof pitch (used by builders/roofers) is a slope.

Roof with rise 6 and run 12. — **Figure 3.4J** (Image by Home Inspection Insider)

Example 3.4.6

A roof rises 6 feet for every 12 feet of horizontal distance.
Find the slope of the roof.

Solution

Draw and label a diagram (not drawn to scale).

Diagram of a roof with a rise of 6 feet and a horizontal run of 12 feet. — **Figure 3.4K**

The slope of the roof is
$m = \frac{6}{12} = \frac{1}{2}$ .

Note: In construction, roofers usually describe this as a
6-in-12 pitch rather than reducing the fraction.
In this course, we will typically reduce slopes to simplest form.

Highway Grade

Highway grade is another real-world application of slope and is usually expressed as a
percent. While driving on the interstate, you may see a warning sign like the one below.
These signs are especially important for trucks, RVs, and drivers in poor weather conditions.

Roadside sign warning of a 7% steep grade ahead. — **Figure 3.4L** (“Steep Grade Ahead: 7 percent” by formulanone licensed CC BY-NC-SA 2.0)

A 7% grade means that the road rises (or falls) 7 feet for every 100 feet of horizontal distance.
In other words, the slope of the road is
$\frac{7}{100} = 0.07$ .

Diagram showing a road with a rise of 7 feet over a horizontal run of 100 feet. — **Figure 3.4M**

Example 3.4.7

A highway rises 300 feet over a horizontal distance of 2 miles.
Find the slope of the highway and express it as a percent grade.

Solution

First, convert miles to feet so that the units match.
There are 5,280 feet in 1 mile.

$2\text{ miles} = 10,\!560\text{ feet}$

Diagram showing a rise of 300 feet over a horizontal run of 10,560 feet. — **Figure 3.4N**

The slope of the highway is
$m = \frac{300}{10,\!560} \approx 0.028$ .

To express this as a percent grade:
$0.028 \times 100\% = 2.8\%$ .

So, the highway has a 2.8% grade.

Wheelchair Ramp

To ensure accessibility, wheelchair ramps must not be too steep.
According to ADA (Americans with Disabilities Act) guidelines, the maximum allowable slope for a wheelchair ramp is:

1 unit of rise for every 12 units of horizontal run
(often written as 1:12)

This means the ramp should rise no more than 1 inch for every 12 inches of horizontal distance.

Wheel chair ramp showing 1:12 slope — **Figure 3.4O** (“WheelChairRamp” by Reliable Ramps. Used with permission.)

Example 3.4.8

A wheelchair ramp rises 2 feet over a horizontal distance of 20 feet.

Find the slope of the ramp.
Does this ramp meet ADA guidelines? Explain.

Solution

A. Draw and label a diagram (not drawn to scale).

Diagram of a wheelchair ramp with a rise of 2 feet and a run of 20 feet. — **Figure 3.4P**

The slope of the ramp is
$m = \frac{2}{20} = \frac{1}{10} = 0.1$ .

B. ADA guidelines recommend a maximum slope of
$\frac{1}{12} \approx 0.083$ .

Since
$\frac{1}{10} > \frac{1}{12}$
(or $0.1 > 0.083$ ),
the ramp is too steep and does not meet ADA standards.

Try It

Finding Slope from an Equation

When an equation is written in slope-intercept form:

$y=mx+b$

the slope is the coefficient $m$ .

Example 3.4.9 — Slope from Standard Form

Find the slope of $9x+3y=6$ .

Solution

Solve for $y$ :

$9x+3y=6$

$3y=-9x+6$

$y=-3x+2$

Slope: $-3$

Example 3.4.10 — Slope with a Negative y

Find the slope of $2x-y=5$ .

Solution

$2x-y=5$

$-y=-2x+5$

$y=2x-5$

Slope: $2$

Try It

Parallel and Perpendicular Lines

The slope of a line tells us more than just how steep the line is. It also allows us to describe how two lines are
related to each other. Two of the most important relationships are parallel and perpendicular lines.

Parallel lines run in the same direction and never intersect.
Because they rise and run at the same rate, parallel lines always have the
same slope, even though they may have different y-intercepts.

Parallel lines with the same slope of 2/3 — **Figure 3.4Q**

Perpendicular lines intersect at a right angle (90°).
This happens when the slopes of the two lines are
negative reciprocals of each other — meaning their slopes multiply to
$-1$ .

Two lines perpendicular (90 deg) with slopes that are negative reciprocals: y = (-2/3)x + 4 and y = (3/2)x -2 — **Figure 3.4R**

These relationships are especially useful in geometry, construction, road design,
and graphing systems where right angles and consistent spacing matter.

**Table 3.4B**
Relationship	Description	Example
Parallel	Same slope	$y = 3x + 2$ and $y = 3x - 5$
Perpendicular	Slopes are negative reciprocals	$y = 2x + 4$ and $y = -\frac{1}{2}x + 7$

Key takeaway:
Same slope → parallel
Negative reciprocals → perpendicular

Example 3.4.11 — Parallel, Perpendicular, or Neither

Determine whether each pair of lines is parallel, perpendicular, or neither.

1. $y = 3x + 2$ and $y = 3x - 5$

The slopes of both lines are $3$ .

Since the slopes are equal, the lines are parallel.

2. $y = -2x + 4$ and $y = \frac{1}{2}x - 7$

The slopes are $-2$ and $\frac{1}{2}$ .

Because the slopes are negative reciprocals, the lines are perpendicular.

3. $y = \frac{2}{3}x + 1$ and $y = \frac{2}{3}x - 6$

The slopes of both lines are $\frac{2}{3}$ .

Since the slopes are equal, the lines are parallel.

4. $y = \frac{3}{2}x + 6$ and $y = \frac{2}{3}x - 3$

The slopes are $\frac{3}{2}$ and $\frac{2}{3}$ .

The slopes are not equal and are not negative reciprocals.

The lines are neither parallel nor perpendicular.

Try It

Conceptual Review

**Table 3.4C**
Concept	Key Idea
Slope formula	$m=\frac{y_2-y_1}{x_2-x_1}$
Horizontal line	$y=\text{constant}$ → slope $0$
Vertical line	$x=\text{constant}$ → undefined slope
Positive slope	Line rises from left to right
Negative slope	Line falls from left to right
Parallel lines	Same slope
Perpendicular lines	Slopes are negative reciprocals

Try It

Key Takeaways

Slope describes steepness and direction.
Use the slope formula when given two points.
Use slope-intercept form to identify slope directly.
Parallel lines have equal slopes.
Perpendicular lines have slopes that are negative reciprocals.
Slope represents real-world rate of change.

License

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Introduction

Finding Slope from a Graph

Summary

Finding Slope Between Two Points

Slope in Real-World Contexts

Roof Pitch

Highway Grade

Wheelchair Ramp

Finding Slope from an Equation

Parallel and Perpendicular Lines

Conceptual Review

License

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