Lesson 2.2: Linear Functions: Properties & Graphs

Slope, Intercepts, and Real Meanings

Explore y = mx + b. Interpret m as unit rate and b as initial value. Understand how average rate on any interval equals the slope.

Learning Objectives

Interpret slope as unit rate
Find y-intercept and x-intercept
Compute average rate of change
Model linear contexts and piecewise fees

Form & Meanings

Linear form: $y=mx+b$ . $m$ is slope (rise/run), $b$ is y-intercept (value at $x=0$ ).

Average rate on $[x_1, x_2]$ : $\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ equals $m$ for linear functions.

Other common forms: Point-slope $y-y_1=m(x-x_1)$ ; Standard $Ax+By=C$ (with $A, B, C$ constants). Convert between forms to match context.

Parallel & perpendicular: Lines with the same slope are parallel; if slopes are $m_1$ and $m_2$ , then perpendicular when $m_1cdot m_2=-1$ (non-vertical).

Advanced Worked Examples

A) Model from two points

Find the line through $(2,5)$ and $(7,20)$ in slope-intercept form.

$m=\dfrac{20-5}{7-2}=\dfrac{15}{5}=3$

Use point $(2,5)$ in $y=mx+b$ → $5=3\cdot 2+b\Rightarrow b=-1$

Line: $y=3x-1$

B) Parallel / Perpendicular

Find the line through $(4,-2)$ that is perpendicular to $y=\tfrac{1}{2}x+3$ .

Slope of given line $m_1=\tfrac{1}{2}$ → perpendicular slope $m_2=-2$

Point-slope: $y+2=-2(x-4)\Rightarrow y=-2x+6$

C) Standard/Intercept form conversion

Convert $2x-3y=12$ to slope-intercept and find intercepts.

$-3y=-2x+12\Rightarrow y=\tfrac{2}{3}x-4$

y-intercept: $(0,-4)$ ; x-intercept: set $y=0$ → $2x=12\Rightarrow x=6$ → $(6,0)$

D) Piecewise fee with threshold

Gym charges $30 per month plus $5 per class after the first 4 free classes per month. Let $x$ be classes taken (nonnegative integer). Write a piecewise $y(x)$ .

$y(x)=\begin{cases}30,& 0\le x\le 4\\ 30+5(x-4),& x>4\end{cases}$

Average rate for $x>4$ equals 5; base fee reflected in intercept shift.

Common Pitfalls

Confusing average rate with instantaneous rate: for linear functions they match; for non-linear they differ.
Mixing up x- and y-intercepts: x-intercept sets $y=0$ , y-intercept sets $x=0$ .
For perpendicular lines using reciprocal instead of negative reciprocal; must satisfy $m_1m_2=-1$ .
Dropping parentheses when using point-slope: write $y-y_1=m(x-x_1)$ , not $y-y_1=mx-x_1$ .

Worked Example (Taxi Pricing)

Base fare 8 within 3 km, then 2 per km after. Write y(x) for x ≥ 3 and find y(5).

$y=8+2(x-3)=2x+2$

At $x=5$ → $y=12$

Slope $m=2$ means +2 per extra km; intercept $b=2$ is theoretical value at $x=0$

Practice

1) Find equation of line through $(3,-1)$ parallel to $y=4x+5$ .

Solution

Same slope 4 → point-slope: $y+1=4(x-3)$ → $y=4x-13$

2) Line with x-intercept 6 and y-intercept -3: write in standard form.

Solution

Intercept form $\tfrac{x}{6}+\tfrac{y}{-3}=1$ → $x-2y=6$

3) A service costs a setup fee of $15 and $0.8 per mile. Express cost vs. miles and find average rate over [10, 25].

Solution

$y=0.8x+15$ ; linear → average rate = slope = $0.8$

4) Through $(1,2)$ and perpendicular to $2x+3y=6$ .

Solution

Given slope $m_1=-\tfrac{2}{3}$ → perpendicular $m_2=\tfrac{3}{2}$ ; point-slope $y-2=\tfrac{3}{2}(x-1)$ → $y=\tfrac{3}{2}x+\tfrac{1}{2}$

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