The relationship between two positive variables and is modeled by a power function of the form for . Replacing with multiplies by . If when , which equation represents this model?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 171 | Free SAT Prep | Aniko

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Question 171·Hard·Nonlinear Functions

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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?

For power function modeling questions of the form $y = ax^k$ , first use the scaling information (how $y$ changes when $x$ is multiplied by a constant) to set up an equation like $c^k =$ given factor and solve for $k$ using exponent rules and common bases such as powers of $2$ or $3$ . Once $k$ is known, plug a given $(x,y)$ point into $y = ax^k$ to solve for $a$ . Finally, write the full function and compare it to the answer choices, and as a quick check make sure it satisfies all conditions in the problem (both the scaling behavior and any specific points).

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Express y when x is replaced by 9x

Start with $y = ax^k$ . If you replace $x$ with $9x$ , what expression do you get for the new $y$ in terms of $a$ , $k$ , and $x$ ?

Relate 9^k to 27

When you replace $x$ with $9x$ , the new $y$ becomes $9^k$ times the original $y$ . Set $9^k$ equal to $27$ , then rewrite $9$ and $27$ as powers of $3$ to solve for $k$ .

Use the point (4, 24) to find a

Once you know $k$ , plug $x=4$ and $y=24$ into $y = ax^k$ to form an equation and solve for $a$ .

Check both conditions against the choices

Any correct answer must satisfy both: (1) replacing $x$ with $9x$ multiplies $y$ by $27$ , and (2) $y=24$ when $x=4$ . Test each choice against these two facts.

Desmos Guide

Define functions for each answer choice

In Desmos, enter four functions:

fA(x) = 3x^(3/2)
fB(x) = 3x^(2/3)
fC(x) = 12*sqrt(x)
fD(x) = 6x^(3/2) These correspond to choices A, B, C, and D.

Test the scaling condition for each function

In new lines, compute the ratio when $x$ is replaced by $9x$ for a convenient positive value, such as $x=1$ :

fA(9)/fA(1)
fB(9)/fB(1)
fC(9)/fC(1)
fD(9)/fD(1) Only the correct model will give a ratio of exactly 27.

Test the point (4, 24) for the remaining candidates

For the function(s) that passed the scaling test, evaluate at x = 4:

fA(4)
fB(4)
fC(4)
fD(4) The correct choice is the one whose value at x = 4 equals 24 and also satisfied the scaling condition in the previous step.

Step-by-step Explanation

Translate the scaling information into an equation for k

We are told that $y = ax^k$ for $x>0$ , and that replacing $x$ with $9x$ multiplies $y$ by $27$ .

Start from the model:

Original: $y = ax^k$
Replace $x$ with $9x$ : $y_{new} = a(9x)^k = a \cdot 9^k x^k$

So the new $y$ is $9^k$ times the old $y$ . The problem says this factor is $27$ , so:

9^k = 27

This equation will let us solve for $k$ .

Solve for the exponent k using powers of 3

Rewrite $9$ and $27$ as powers of $3$ :

$9 = 3^2$
$27 = 3^3$

Substitute into $9^k = 27$ :

(3^2)^k = 3^3

Using the rule $(a^m)^n = a^{mn}$ gives:

3^{2k} = 3^3

Since the bases are the same and positive, the exponents must be equal:

2k = 3 \quad \Rightarrow \quad k = \frac{3}{2}

So the model has the form $y = ax^{3/2}$ .

Use the given point to find the coefficient a

We are told that $y = 24$ when $x = 4$ . Substitute into $y = ax^{3/2}$ :

24 = a \cdot 4^{3/2}

Compute $4^{3/2}$ by thinking of it as $(\sqrt{4})^3$ :

$\sqrt{4} = 2$
$2^3 = 8$ , so $4^{3/2} = 8$

Now solve for $a$ :

24 = 8a \quad \Rightarrow \quad a = 3

Match the model to the answer choices

From the previous steps, the function is $y = 3x^{3/2}$ .

Compare with the answer choices:

A) $y = 3x^{3/2}$
B) $y = 3x^{2/3}$
C) $y = 12\sqrt{x}$
D) $y = 6x^{3/2}$

The model exactly matches choice A) $y = 3x^{3/2}$ , so A is correct.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation