Lesson 4.4: Understand Slope of a Line

New Concept

Slope, $m$, measures a line's steepness and direction. You'll learn to calculate this 'tilt' as the ratio of vertical change (rise) to horizontal change (run), using graphs ($m = \dfrac{\operatorname{rise}}{\operatorname{run}}$) and coordinates ($m = \dfrac{y_2 - y_1}{x_2 - x_1}$).

What’s next

You're ready to start! Up next are interactive examples and practice cards where you'll find the slope of any line from its graph or given points.

Property

The slope of a line of a line is $m = \frac{\operatorname{rise}}{\operatorname{run}}$. The rise measures the vertical change and the run measures the horizontal change between two points on the line.

Examples

• A line rises 4 units for every 5 units it runs horizontally. Its slope is $m = \frac{4}{5}$.
• A line drops 2 units for every 3 units it runs horizontally. Its rise is $-2$, so its slope is $m = \frac{-2}{3}$ or $-\frac{2}{3}$.
• On a geoboard, if you connect two pegs by going up 3 units and right 2 units, the slope of the line is $m = \frac{3}{2}$.

Explanation

Slope tells you how steep a line is. Think of it as 'rise over run'. For every number of units you move horizontally (run), the slope tells you how many units you move vertically (rise).

Property

To find the slope of a line from its graph:

1. Locate two points on the graph 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, $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A line passes through $(1, 2)$ and $(4, 8)$. The rise is $8 - 2 = 6$ and the run is $4 - 1 = 3$. The slope is $m = \frac{6}{3} = 2$.
• A line passes through $(0, 6)$ and $(2, 2)$. The rise is $2 - 6 = -4$ and the run is $2 - 0 = 2$. The slope is $m = \frac{-4}{2} = -2$.
• A line on a graph connects points $(-3, 1)$ and $(5, 7)$. The rise is $7-1=6$ and the run is $5-(-3)=8$. The slope is $m = \frac{6}{8} = \frac{3}{4}$.

Explanation

To find slope from a graph, pick two easy-to-read points. Count the vertical distance (rise) and horizontal distance (run) to get from one point to the other. The slope is simply the rise divided by the run.

Property

The slope of a horizontal line, $y = b$, is 0. The slope of a vertical line, $x = a$, is undefined.

Examples

• The line $y = 5$ is a horizontal line. Its slope is 0.
• The line $x = -3$ is a vertical line. Its slope is undefined.
• A line passing through the points $(2, 6)$ and $(9, 6)$ has a rise of $6 - 6 = 0$, so its slope is $m = \frac{0}{7} = 0$.

Explanation

A horizontal line is perfectly flat, so its 'rise' is always zero, making the slope 0. A vertical line is infinitely steep; its 'run' is zero, and since we can't divide by zero, its slope is undefined.

Property

The slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$ is $m = \frac{y_2 - y_1}{x_2 - x_1}$. This is the slope formula. The slope is the difference in the y-coordinates divided by the difference in the x-coordinates.

Examples

• For the points $(2, 5)$ and $(4, 11)$, the slope is $m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$.
• For the points $(-3, 6)$ and $(1, 4)$, the slope is $m = \frac{4 - 6}{1 - (-3)} = \frac{-2}{4} = -\frac{1}{2}$.
• For the points $(5, -1)$ and $(-2, 3)$, the slope is $m = \frac{3 - (-1)}{-2 - 5} = \frac{4}{-7} = -\frac{4}{7}$.

Explanation

The slope formula is a tool to find a line's steepness without a graph. It calculates the rise by subtracting y-values ($y_2 - y_1$) and the run by subtracting x-values ($x_2 - x_1$), then divides them.

Property

To graph a line given a point and the slope:

1. Plot the given point.
2. Use the slope formula $m = \frac{\operatorname{rise}}{\operatorname{run}}$ to identify the rise and the run.
3. Starting at the given point, count out the rise and run to mark the second point.
4. Connect the points with a line.

Examples

• To graph a line through $(1, 2)$ with slope $m = \frac{3}{5}$, start at $(1, 2)$, move up 3 units and right 5 units to plot a second point at $(6, 5)$.
• To graph a line through $(-2, 4)$ with slope $m = -3$, treat the slope as $\frac{-3}{1}$. From $(-2, 4)$, move down 3 units and right 1 unit to plot a second point at $(-1, 1)$.
• To graph a line through the y-intercept $(0, -1)$ with slope $m = \frac{2}{3}$, start at $(0, -1)$, move up 2 units and right 3 units to plot a second point at $(3, 1)$.

Explanation

Think of it as 'point and directions'. Start by plotting the given point. Then use the slope's rise and run as steps to find a second point. Connect them to draw your line.

Property

The 'pitch' of a building’s roof is the slope of the roof. The grade of a road or the slope of a drainage pipe are other real-world applications of slope. We use the formula $m = \frac{\operatorname{rise}}{\operatorname{run}}$ to calculate these values.

Examples

• A roof rises 6 feet for every 18 feet of horizontal run. The pitch of the roof is its slope: $m = \frac{6}{18} = \frac{1}{3}$.
• A road drops 4 feet for every 100 feet of horizontal distance. The grade of the road is $m = \frac{-4}{100} = -\frac{1}{25}$.
• A wheelchair ramp needs to rise 2 feet. If the required slope is $\frac{1}{12}$, the horizontal run must be 24 feet, since $\frac{2}{24} = \frac{1}{12}$.

Explanation

Slope is everywhere in the real world. It describes the steepness of hills, the pitch of roofs, and the angle of wheelchair ramps. A positive slope goes up, and a negative slope goes down.