Slope and Linear Functions

What Is Slope?

Slope measures the steepness of a line — the rate of change.

$m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{\Delta y}{\Delta x}$

Types of Slope

| Type | Description | Example | |------|-------------|---------| | Positive | Line goes up left to right | $m = 2$ | | Negative | Line goes down left to right | $m = -3$ | | Zero | Horizontal line | $m = 0$ | | Undefined | Vertical line | |

Slope-Intercept Form

$y = mx + b$

• $m$ = slope (rate of change)
• $b$ = y-intercept (where the line crosses the y-axis)

Example: $y = 2x + 3$

• Slope = 2 (goes up 2 for every 1 right)
• Y-intercept = 3 (crosses y-axis at $(0, 3)$)

Graphing a Line

From $y = -\frac{1}{2}x + 4$:

1. Plot the y-intercept $(0, 4)$
2. Use slope: down 1, right 2 → plot $(2, 3)$
3. Connect the points

Finding Slope from a Graph

Pick two points on the line and use the slope formula: Points: $(1, 3)$ and $(4, 9)$ $m = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

Point-Slope Form

$y - y_1 = m(x - x_1)$

Through $(2, 5)$ with slope 3: $y - 5 = 3(x - 2)$ $y = 3x - 1$

Parallel and Perpendicular Lines

• Parallel lines: Same slope ($m_1 = m_2$)
• Perpendicular lines: Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$)

Real-world slope: A roof has a pitch of 6:12 (rise of 6 for every run of 12), meaning $m = \frac{1}{2}$.