The function is exponential and can be written in the form with and . If and , which equation defines ?

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SAT Nonlinear Functions Question 42 | Free SAT Prep | Aniko

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Question 42·Hard·Nonlinear Functions

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The function $g$ is exponential and can be written in the form $g(x)=ab^x$ with $a>0$ and $b>0$ . If $g(3)=120$ and $g(7)=30$ , which equation defines $g$ ?

For exponential questions with two given points, write the function in the shifted form $g(x)=g(x_0)\,b^{x-x_0}$ using one of the points. Then plug in the second point to find $b^{\Delta x}$ as the ratio of the outputs, where $\Delta x$ is the change in $x$ between the points. Finally, use exponent rules to express $b^{x-x_0}$ in terms of that known power, which typically yields a base you recognize (like $1/4$ ) and an exponent involving $(x-x_0)$ , allowing you to match the correct answer choice quickly.

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Use the two given points to compare outputs

You know $g(3)=120$ and $g(7)=30$ , and the function is exponential. Think about how an exponential function changes when $x$ increases by a fixed amount, like from 3 to 7.

Relate the ratio of outputs to a power of b

Write $g(x)$ in the form $g(x)=120b^{x-3}$ using the fact that $g(3)=120$ . Then plug in $x=7$ and use $g(7)=30$ to find an equation involving $b^4$ .

Turn b⁴ into a base you recognize

Once you know $b^4 = 1/4$ , use exponent rules to rewrite $b^{x-3}$ in terms of $b^4$ so that the base becomes $1/4$ and the exponent involves $(x-3)/4$ .

Desmos Guide

Graph each answer choice

In Desmos, enter each option as a separate equation, for example:

Option A: y1 = 120*(1/4)^((x-3)/4)
Option B: y2 = 120*(1/4)^(x-3)
Option C: y3 = 120*(1/2)^(x-3)
Option D: y4 = 120*(4)^((x-3)/4) This will show four different curves on the same axes.

Check which graphs pass through the given points

For each equation (say $y_1$ ), click the gear icon and add a table. In the table, enter $x=3$ and $x=7$ and read off the corresponding $y$ values. Repeat for $y_2$ , $y_3$ , and $y_4$ .

Match the function to the conditions

Identify which equation gives $y=120$ when $x=3$ and $y=30$ when $x=7$ . The equation whose graph hits both of these points is the correct choice.

Step-by-step Explanation

Write g in a useful exponential form using x = 3

We are told $g$ is exponential and can be written as $g(x)=ab^x$ .

Because we know $g(3)$ , it is often easier to write the function in shifted form around $x=3$ :

g(x) = g(3) \, b^{x-3}

We are given $g(3)=120$ , so

g(x) = 120 \, b^{x-3}.

Now we only need to figure out the base $b$ .

Use g(7)=30 to find b⁴

Use the point $(7,30)$ in the equation $g(x)=120b^{x-3}$ .

At $x=7$ :

g(7) = 120 \, b^{7-3} = 120 \, b^4.

But $g(7)=30$ , so

30 = 120 \, b^4.

Divide both sides by 120:

b^4 = \frac{30}{120} = \frac{1}{4}.

So the fourth power of $b$ is $1/4$ .

Rewrite b^(x−3) using the fact that b⁴ = 1/4

We know $g(x)=120b^{x-3}$ and $b^4=1/4$ .

Use exponent rules to rewrite $b^{x-3}$ in terms of $b^4$ :

b^{x-3} = \bigl(b^4\bigr)^{\frac{x-3}{4}}.

Substitute $b^4 = 1/4$ :

b^{x-3} = \left(\frac{1}{4}\right)^{\frac{x-3}{4}}.

Plug this back into $g(x)$ :

g(x) = 120 \left(\frac{1}{4}\right)^{\frac{x-3}{4}}.

So the equation that defines $g$ is $g(x)=120(1/4)^{(x-3)/4}$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation