Slope-Intercept Form Preview
Introduce y = mx + b as the slope-intercept form of a line. Identify slope (m) and y-intercept (b) from an equation, and graph lines quickly using these two values.
A lesson of Pre-Algebra, in The Coordinate Plane & Linear Relationships, about 10 minutes.
Slope-Intercept Form
By the end of this lesson, you will be able to identify slope and y-intercept from an equation, write equations of lines, and interpret them in context.
Slope-Intercept Form
$$y = mx + b$$
- $m$ = slope (rate of change)
- $b$ = y-intercept (value of y when $x = 0$)
This form makes it easy to graph a line and read its properties immediately.
Reading the Equation
Example: $y = 3x - 5$
- Slope $m = 3$: for every 1 unit increase in x, y increases by 3
- y-intercept $b = -5$: the line crosses the y-axis at $(0, -5)$
Example: $y = -\frac{2}{3}x + 4$
- Slope $m = -2/3$: for every 3 units right, y decreases by 2
- y-intercept $b = 4$: the line crosses at $(0, 4)$
Writing Equations from Information
Given slope and y-intercept: substitute directly into $y = mx + b$.
$m = 2$, $b = -1$ → $y = 2x - 1$
Given slope and one point: substitute the point to find $b$.
$m = 3$, passes through $(2, 7)$:
$7 = 3(2) + b$ → $b = 7 - 6 = 1$ → $y = 3x + 1$
Given two points: find slope first, then y-intercept.
Through $(1, 4)$ and $(3, 10)$:
$m = \frac{10-4}{3-1} = 3$; then $4 = 3(1) + b$ → $b = 1$ → $y = 3x + 1$
Converting Standard Form to Slope-Intercept
Example: $4x + 2y = 10$
Solve for y: $2y = -4x + 10$ → $y = -2x + 5$
Slope = $-2$; y-intercept = $5$.
Real-World Interpretation
Example: A taxi charges a $$3$ base fee plus $$2.50$ per mile.
Cost equation: $y = 2.50x + 3$
- Slope 2.50: each additional mile costs $$2.50$
- y-intercept 3: the starting cost before any miles
Summary
$y = mx + b$: slope $m$ = rate of change; y-intercept $b$ = starting value at $x = 0$.
Writing equations:
- Slope + y-intercept: plug in directly