When y=kxn, multiplying x by a factor r multiplies y by rn. The constant k cancels, so the change in y can be found without knowing k. Why: new y=k(rx)n=rn(kxn)=rn x original y. **Change in x** | **Multiplier for x** | **y=kx** | **y=kx2** | **y=kx3** doubled | 2 | x 2 | x 4 | x 8 halved | 0.5 | x 0.5 | x 0.25 | x 0.125 increased by 50% | 1.5 | x 1.5 | x 2.25 | x 3.375 decreased by 20% | 0.8 | x 0.8 | x 0.64 | x 0.512 Method - Convert the change in x to a multiplier: an increase of 30% gives 1.3; a decrease of 30% gives 0.7. - Raise the multiplier to the power in the equation (use a square root for sqrt(x)). - Convert the result back: percentage change in y=(multiplier-1) x 100%. A negative answer is a decrease. Example The power, P watts, produced by a wind turbine is directly proportional to the cube of the wind speed, v m/s. The wind speed increases by 20%. Find the percentage increase in P. Multiplier for v: 1.2 Multiplier for P: 1.23=1.728 Increase: (1.728-1) x 100%=72.8% Check: a 20% increase in v does not give a 60% increase in P; the multiplier is cubed, not the percentage. Test with numbers: v=10 gives P=1000k; v=12 gives P=1728k, which is 72.8% more.

Changes in direct proportion

Secondary 2

When $y=kx^n$, multiplying $x$ by a factor $r$ multiplies $y$ by $r^n$. The constant $k$ cancels, so the change in $y$ can be found without knowing $k$.

Why: new $y=k(rx)^n=r^n\left(kx^n\right)=r^n\times\text{original}\hspace{0.5em}y$.

Change in $x$	Multiplier for $x$	$y=kx$	$y=kx^2$	$y=kx^3$
doubled	$2$	$\times2$	$\times4$	$\times8$
halved	$0.5$	$\times0.5$	$\times0.25$	$\times0.125$
increased by $50\%$	$1.5$	$\times1.5$	$\times2.25$	$\times3.375$
decreased by $20\%$	$0.8$	$\times0.8$	$\times0.64$	$\times0.512$

Method

Convert the change in $x$ to a multiplier: an increase of $30\%$ gives $1.3$; a decrease of $30\%$ gives $0.7$.
Raise the multiplier to the power in the equation (use a square root for $\sqrt{x}$).
Convert the result back: percentage change in $y=(\text{multiplier}\hspace{0.5em}-1)\times100\%$. A negative answer is a decrease.

Example

The power, $P$ watts, produced by a wind turbine is directly proportional to the cube of the wind speed, $v$ m/s. The wind speed increases by $20\%$. Find the percentage increase in $P$.

Multiplier for $v$: $1.2$

Multiplier for $P$: $1.2^3=1.728$

Increase: $(1.728-1)\times100\%=72.8\%$

Check: a $20\%$ increase in $v$ does not give a $60\%$ increase in $P$; the multiplier is cubed, not the percentage. Test with numbers: $v=10$ gives $P=1000k$; $v=12$ gives $P=1728k$, which is $72.8\%$ more.

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Change in \(x\)	Multiplier for \(x\)	\(y=kx\)	\(y=kx^2\)	\(y=kx^3\)
doubled	\(2\)	\(\times2\)	\(\times4\)	\(\times8\)
halved	\(0.5\)	\(\times0.5\)	\(\times0.25\)	\(\times0.125\)
increased by \(50\%\)	\(1.5\)	\(\times1.5\)	\(\times2.25\)	\(\times3.375\)
decreased by \(20\%\)	\(0.8\)	\(\times0.8\)	\(\times0.64\)	\(\times0.512\)