Linear Graphs, Slope, and the Slope-Intercept Form

Linear Graphs, Slope, and the Slope-Intercept Form

Understanding coordinate planes and linear graphs is essential for ~55% of the algebraic section.

1. The Slope Formula

The slope $m$ measures the steepness and direction of a line: $$m = \frac{\text{Rise}}{\text{Run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

2. Slope-Intercept Form

$$y = mx + b$$

• $m$ = slope (rate of change)
• $b$ = $y$-intercept (the value of $y$ where the line crosses the $y$-axis at $(0, b)$)

3. Parallel and Perpendicular Lines

• Parallel lines have identical slopes: $$m_1 = m_2$$
• Perpendicular lines have negative reciprocal slopes: $$m_1 \cdot m_2 = -1$$ (e.g., $2$ and $-\frac{1}{2}$).

Worked Example:

Find the equation of the line passing through $(2, 5)$ and $(4, 11)$:

1. Find slope: $$m = \frac{11 - 5}{4 - 2} = \frac{6}{2} = 3$$
2. Use $y = mx + b$ with point $(2, 5)$: $$5 = 3(2) + b \implies 5 = 6 + b \implies b = -1$$
3. Equation: $$y = 3x - 1$$