Question 682·Hard·Math

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The relationship between two positive variables $x$ and $y$ is modeled by a power function of the form $y = a x^k$ for $x > 0$ . Replacing $x$ with $9x$ multiplies $y$ by $27$ . If $y = 24$ when $x = 4$ , which equation represents this model?

A power function raises the input to a fixed exponent. When the input is multiplied by a number, the output is multiplied by that number raised to the exponent. Use this input-output multiplier to find the exponent, then use the known input-output pair to find the coefficient, the number in front. Don’t mistake the input multiplier for the output multiplier.

Hints

Hint 1
In a power function $y=ax^k$ , replacing $x$ with $9x$ turns $x^k$ into $(9x)^k=9^kx^k$ . The coefficient $a$ stays the same. What factor multiplies the whole output?
Hint 2
A regression lets Desmos find a number that makes an equation true. Set the output factor you found equal to $27$ , then replace $=$ with $\sim$ to find the exponent $k$ .
Hint 3
Once you have the exponent, put the known input and output into $y=ax^k$ . That leaves only the coefficient $a$ to find. In a second Desmos regression, type the exponent’s numerical value rather than its letter.

Step-by-step

Find the exponent from the multiplier
Step 1
Find the output factor in the model
Write $y(t)$ for the output at input $t$ . Replace $x$ with $9x$ in the power function:
$y(9x)=a(9x)^k.$
Give the exponent to both factors of $9x$ :
$y(9x)=a9^kx^k.$
Since $ax^k=y(x)$ , rewrite:
$y(9x)=9^ky(x).$
So the output factor is $9^k$ , not $9k$ .
Step 2
Use the output change to find the exponent
The words “multiplies $y$ by $27$ ” mean $y(9x)=27y(x)$ . Match this with the output from step 1:
$9^ky(x)=27y(x).$
Divide by $y(x)$ , which is positive:
$9^k=27.$
Use a regression: type $9^k\sim27$ . Under PARAMETERS, Desmos shows $k=1.5$ , which is $\frac32$ .
Step 3
Use the known pair to find the coefficient
Put $x=4$ and $y=24$ into the model, using $k=\frac32$ :
$24=a\cdot4^{3/2}.$
Type $a\cdot4^{1.5}\sim24$ . Desmos shows $a=3$ under PARAMETERS. Type $1.5$ , not $k$ , because another regression would fit $k$ again.
Step 4
Write the equation
Put the coefficient and exponent into the given form. The equation that meets both conditions is
$y=3x^{3/2}.$
Choice D.

Question 682·Hard·Math

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