Section 2.2: Slope & Intercepts
The slope of a line measures its steepness and direction. For a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \(m\) is:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
The y-intercept is where the line crosses the y-axis (\(x=0\)), and the x-intercept is where it crosses the x-axis (\(y=0\)).
Key Points:
- Positive slope → line rises left to right
- Negative slope → line falls left to right
- Zero slope → horizontal line
- Undefined slope → vertical line
Example 1
Find the slope of the line through points (2,3) and (5,11).
Use the slope formula:
\( m = \frac{11 - 3}{5 - 2} = \frac{8}{3} \).
Slope = \( \frac{8}{3} \)
Example 2
Find the x- and y-intercepts of \( y = 2x - 4 \).
y-intercept: \( x = 0 \) → \( y = 2(0) - 4 = -4 \) → (0,-4)
x-intercept: \( y = 0 \) → \( 0 = 2x - 4 \) → \( x = 2 \) → (2,0)
Practice Problems
- Find the slope of the line through (1,2) and (4,8).
- Determine the slope of the line through (-2,5) and (3,-10).
- Find the y-intercept of \( y = -3x + 7 \).
- Find the x-intercept of \( 4x + 2y = 8 \).
- Identify the slope and intercepts for \( y = 0.5x - 1 \).