Slope and Slope-Intercept Form
Slope
The slope of a line measures its steepness — how much the line rises (or falls) for each unit of horizontal movement. Slope is the ratio of vertical change to horizontal change, often described as "rise over run," and it is one of the most important concepts in all of mathematics.
The slope-intercept form of a line, y = mx + b, elegantly packages two pieces of information: the slope m (the rate of change) and the y-intercept b (the starting value). This form makes graphing fast and interpretation intuitive.
Understanding slope prepares you for rates of change in science, marginal cost in economics, and eventually the derivative in calculus.
The slope of a line through $(x_1, y_1)$ and $(x_2, y_2)$ is:
$$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$
| Slope | Direction |
|---|---|
| $m > 0$ | Rises left to right ↗ |
| $m < 0$ | Falls left to right ↘ |
| $m = 0$ | Horizontal — |
| Undefined | Vertical | |
Slope through $(2, 3)$ and $(5, 9)$.
$$m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$$
Slope through $(-1, 4)$ and $(3, -2)$.
$$m = \frac{-2 - 4}{3 - (-1)} = \frac{-6}{4} = -\frac{3}{2}$$
Slope-Intercept Form
$$y = mx + b$$
$m$ = slope, $b$ = $y$-intercept (the point $(0, b)$).
Identify slope and $y$-intercept of $y = -\dfrac{2}{3}x + 4$.
Slope $= -\dfrac{2}{3}$. $y$-intercept $= (0, 4)$.
Convert $3x + 2y = 12$ to slope-intercept form.
$2y = -3x + 12 \;\Rightarrow\; y = -\dfrac{3}{2}x + 6$. Slope $= -\dfrac{3}{2}$, $y$-int $= 6$.
Practice Problems
Show Answer Key
1. $3$
2. $0$ (horizontal)
3. Undefined (vertical)
4. $m = 5$, $b = -3$
5. $y = 4x - 8$
6. $-3$
7. $y = -\dfrac{2}{5}x + 4$
8. $-\dfrac{3}{2}$
9. $y = \dfrac{1}{3}x - 3$
10. $0$
11. $3(2) - 1 = 5$ ✓ Yes
12. $2$