M151Q Precalculus · Unit 2: Powers, exponentials, logarithms · on Exam 2
MFG 3.1 Variation
Study notes for Variation 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
- Direct variation: $y = kx^n$: $y$ scales with a power of $x$; the graph passes through the origin. Doubling $x$ multiplies $y$ by $2^n$.
- Inverse variation: $y = k/x^n$: as $x$ grows, $y$ shrinks; the graph never touches the axes.
- Find $k$ from one data point, then use the formula. Check a table for variation by testing whether $y/x^n$ (direct) or $yx^n$ (inverse) is constant.
- Scaling arguments: if $y \propto x^2$ and $x$ triples, $y$ becomes $9$ times as large; if $y \propto 1/x^2$, it becomes $1/9$.
Key formulas
Direct / inverse
$$y = kx^n\qquad y = \frac{k}{x^n}$$
Scaling
$$x \to cx \ \Rightarrow\ y \to c^n y\ (\text{direct}),\quad y \to c^{-n}y\ (\text{inverse})$$
Worked example
The weight a beam can support varies inversely with its length. A 4 m beam supports 600 kg. What can a 10 m beam support?
Show the solution
$W = k/L$, $600 = k/4 \Rightarrow k = 2400$. $W(10) = 240$ kg.
Common mistakes
- Treating "varies with" as linear when a power is stated.
- Forgetting that direct variation has no constant term (y = kx, not y = kx + b).
Exam tipWrite the variation equation with k before touching the numbers; then one data point gives k.
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