Direct Variation
Fractions & DecimalsDirect variation describes a relationship where two quantities increase or decrease together at a constant rate, expressed as y = kx.
Formula
y = kx \text{ (}k\text{ is the constant of variation)}
Definition
Direct variation means that when one quantity goes up, the other goes up by the same proportion, and when one goes down, the other goes down too. They move together in the same direction.
Example
If you earn $\$12$ per hour, your total pay varies directly with hours worked: $1$ hr $= \$12$, $2$ hrs $= \$24$, $3$ hrs $= \$36$. Double the hours, double the pay.
Key Insight
"Direct" means they go the same direction together. More hours, more pay. Less hours, less pay. The ratio pay/hours is always $\$12$ - it never changes.
Definition
Direct variation (direct proportion) is a relationship $y = kx$ where $k$ (the constant of variation) is nonzero. Key properties: (1) $y/x = k$ for all nonzero $x$; (2) the graph is a line through the origin with slope $k$; (3) if $x$ doubles, $y$ doubles. To verify direct variation, check that $y/x$ is constant for all data pairs.
Example
Data: $(2, 8)$, $(5, 20)$, $(7, 28)$. Check $y/x$: $8/2=4$, $20/5=4$, $28/7=4$. Constant, so $y=4x$ (direct variation, $k=4$). If one pair were $(5, 21)$, the ratio would be $21/5=4.2$ (different), so it would not be direct variation.
Key Insight
Direct variation graphs always pass through the origin $(0,0)$. If a "linear relationship" does not pass through the origin (like $y=2x+3$), it is linear but not directly proportional. This distinction is crucial in science: Hooke's law ($F=kx$) is direct variation, but temperature scales like Fahrenheit and Celsius are related linearly but not proportionally.
Definition
Direct variation $y = kx$ is a one-dimensional linear map (degree-$1$ homogeneous function): $f(cx) = cf(x)$ for all scalars $c$. It satisfies $f(x+y) = f(x)+f(y)$ (additivity) and $f(cx) = cf(x)$ (homogeneity), making it a linear functional on $\mathbb{R}$. In power law form, $y$ varies directly as $x^n$ means $y = kx^n$ (direct variation of degree $n$), a monomial function.
Example
The period $T$ of a simple pendulum varies directly as the square root of its length $L$: $T = 2\pi\sqrt{L/g}$. This is direct variation with $k = 2\pi/\sqrt{g}$ in the form $T = k\sqrt{L}$. Doubling $L$ multiplies $T$ by $\sqrt{2} \approx 1.414$, not $2$.
Key Insight
Power-law direct variation $y \sim x^n$ appears throughout physics: gravitational force $\sim 1/r^2$ (inverse square), kinetic energy $\sim v^2$, radiation intensity $\sim 1/r^2$. Log-log plots linearize power laws ($\log y = n\log x + \log k$), making the exponent $n$ the slope - a key tool for identifying power-law relationships in experimental data.