Lesson 4: Understand Slope of a Line

New Concept

This lesson introduces slope ($m$), which measures a line's steepness. You'll learn to find it by calculating the ratio of vertical change (rise) to horizontal change (run), using the formula $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

What’s next

Next, you'll master this concept through interactive examples, practice cards, and challenge problems on graphing lines.

Property

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

Examples

• A line goes up 4 units for every 2 units it moves right. The slope is $m = \frac{\operatorname{rise}}{\operatorname{run}} = \frac{4}{2} = 2$.
• On a graph, a line segment goes from a point down 3 units and right 5 units. The slope is $m = \frac{\operatorname{rise}}{\operatorname{run}} = \frac{-3}{5} = -\frac{3}{5}$.
• If a line has a slope of $\frac{1}{3}$, it means for every 1 unit the line goes up, it moves 3 units to the right.

Explanation

Slope tells you how steep a line is. Think of it as 'rise up, run across'. A positive slope means the line goes uphill from left to right, while a negative slope means it goes downhill.

Property

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

Examples

• The slope of the line given by the equation $y = 5$ is $0$.
• A line passes through the points $(1, -4)$ and $(6, -4)$. The rise is $-4 - (-4) = 0$, so the slope is $m = \frac{0}{6-1} = 0$.
• Any line that is parallel to the x-axis has a slope of $0$ because its vertical change is always zero.

Explanation

A horizontal line is perfectly flat, so it has zero rise between any two points. Because the rise is $0$, the slope is always $0$. Think of walking on level ground—there is no slope!

Property

The slope of a vertical line, $x = a$, is undefined.

Examples

• The slope of the line given by the equation $x = 7$ is undefined.
• A line passes through the points $(3, 2)$ and $(3, 9)$. The run is $3 - 3 = 0$. Since division by zero is not possible, the slope is undefined.
• Any line that is parallel to the y-axis has an undefined slope because its horizontal change is always zero.

Explanation

A vertical line goes straight up and down, so its horizontal change, or run, is zero. Since we cannot divide by zero in mathematics, the slope of a vertical line is described as undefined.

Property

To find the slope of the line between two points $(x_1, y_1)$ and $(x_2, y_2)$, use the slope formula:

Examples

• To find the slope of the line between $(2, 3)$ and $(5, 9)$, use the formula: $m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$.
• To find the slope between $(-1, 5)$ and $(3, -3)$, use the formula: $m = \frac{-3 - 5}{3 - (-1)} = \frac{-8}{4} = -2$.
• The slope of the line passing through $(0, 2)$ and $(4, 4)$ is $m = \frac{4 - 2}{4 - 0} = \frac{2}{4} = \frac{1}{2}$.

Explanation

This formula is a shortcut to find a line's slope without needing a graph. It calculates the vertical change (the difference in y-coordinates) and divides it by the horizontal change (the difference in x-coordinates).

Property

To graph a line using a point and slope:

1. Plot the given point.
2. Use the slope formula $m = \frac{\operatorname{rise}}{\operatorname{run}}$ to identify the rise and run.
3. From the plotted point, count the rise and run to find a second point.
4. Connect the two points to draw the line.

Examples

• To graph the line through $(2, 1)$ with slope $m = \frac{2}{5}$, first plot $(2, 1)$. Then, count up 2 units and right 5 units to find the next point at $(7, 3)$ and connect them.
• To graph the line through $(-1, 4)$ with slope $m = -\frac{3}{2}$, plot $(-1, 4)$. Then, count down 3 units and right 2 units to find the point $(1, 1)$ and connect them.
• To graph the line through $(0, -2)$ with slope $m = 3$, write the slope as $\frac{3}{1}$. Plot $(0, -2)$, then count up 3 units and right 1 unit to find the point $(1, 1)$ and connect them.

Explanation

The slope gives you a set of directions to find more points on a line. Start at a known point, then use the rise (up/down) and run (right) to find the next point. It is like a treasure map for your line!

Property

Real-world scenarios like roof pitch, road grade, and accessibility ramps are practical applications of slope. The pitch of a building’s roof is the slope of the roof, calculated as $m = \frac{\operatorname{rise}}{\operatorname{run}}$.

Examples

• A roof has a rise of 8 feet over a horizontal distance (run) of 24 feet. The slope of the roof is $m = \frac{8}{24} = \frac{1}{3}$.
• A wheelchair ramp must have a slope of at most $\frac{1}{12}$. To build a ramp that rises 3 feet (36 inches), the run must be at least $12 \times 36 = 432$ inches, or 36 feet.
• A road has a grade of 5%, which means its slope is $m = \frac{5}{100} = \frac{1}{20}$. This means the road rises 1 foot for every 20 feet of horizontal distance.

Explanation

Slope is a key concept in engineering and construction. It helps determine the safety and function of many things, from a highway's grade to the pitch of a roof designed to shed snow.