M151Q Precalculus · Unit 1: Functions, graphs, models · on Exam 1
MFG 1.4–1.5 Slope, rate of change, and linear functions
Study notes for Slope, rate of change, and linear functions in M151Q Precalculus at Montana State University: the big ideas, key formulas, a worked example, common mistakes and an exam tip. The same topic has endless practice questions and flashcards in Mathub.
Big ideas
- Average rate of change of $f$ on $[a,b]$ is $\dfrac{f(b)-f(a)}{b-a} = \dfrac{\Delta y}{\Delta x}$: the slope of the line through the two points. Units are output units per input unit.
- A linear function has a constant rate of change: $f(x) = b + mx$ (Yoshiwara's order) or $y = mx + b$. $m$ is the slope, $b$ the starting value ($y$-intercept).
- Find a line from two points: slope first, then point–slope $y - y_1 = m(x - x_1)$. From a table: check that equal steps in $x$ give equal steps in $y$; if so it is linear.
- Intercepts of a line: $y$-intercept $(0,b)$, $x$-intercept $(-b/m, 0)$. Parallel lines share slope; perpendicular slopes multiply to $-1$.
- In context: slope tells "how much $y$ changes per one unit of $x$"; the intercept is the value when $x = 0$. Always write the sentence with units.
Key formulas
Average rate of change
$$\frac{\Delta y}{\Delta x} = \frac{f(b)-f(a)}{b-a}$$
Linear function
$$f(x) = b + mx,\qquad m = \frac{y_2-y_1}{x_2-x_1},\qquad y - y_1 = m(x - x_1)$$
Intercepts
$$(0,\,b)\quad\text{and}\quad\left(-\tfrac{b}{m},\,0\right)$$
Worked example
A taxi charges \$3.50 plus \$2.25 per mile. Write $C(m)$, find the cost of 8 miles, and how far \$30 takes you.
Show the solution
$C(m) = 3.50 + 2.25m$. $C(8) = 3.5 + 18 = \$21.50$. $30 = 3.5 + 2.25m \Rightarrow m = 26.5/2.25 \approx 11.8$ miles. The slope 2.25 means each extra mile costs \$2.25.
Common mistakes
- Swapping Δx and Δy in the slope.
- Reporting a rate without units.
- Using y = mx + b with the intercept and slope reversed when the table starts at x ≠ 0.
Exam tipBefore computing a slope from a table, look at the x-spacing. If the steps are unequal, compute Δy/Δx for each pair; a constant ratio is the test for linearity.
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