A recording engineer uses a linear calibration function that converts a dial setting to an output level. The studio software reports an adjusted level defined by In the -plane, the graph of is a line that passes through the points and . Which choice gives an equation for ?

Solution available with step-by-step explanation

SAT Linear Functions Question 72 | Free SAT Prep | Aniko

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Question 72·Hard·Linear Functions

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A recording engineer uses a linear calibration function $f(x)$ that converts a dial setting $x$ to an output level.

The studio software reports an adjusted level $g(x)$ defined by

g(x)=2f(x+3)-f(x-1).

In the $xy$ -plane, the graph of $g$ is a line that passes through the points $(0,3)$ and $(4,27)$ .

Which choice gives an equation for $f$ ?

When $f$ is linear and a new function is built from shifted values like $f(x+c)$ , rewrite $f(x)$ as $mx+b$ and simplify the expression for the new function completely. Then find the equation of the new (graphed or point-defined) line and match slope/intercept (or coefficients of $x$ and the constant term) to solve for $m$ and $b$ .

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Find $g(x)$ first

Use the two given points to find the slope of $g$ , then use one point to find its $y$ -intercept.

Let $f(x)=mx+b$

Because $f$ is linear, write $f(x)$ as $mx+b$ and substitute $x+3$ and $x-1$ into that expression.

Compare coefficients

After simplifying $g(x)=2f(x+3)-f(x-1)$ , match the slope and intercept you get with the slope and intercept of the line through $(0,3)$ and $(4,27)$ .

Desmos Guide

Enter the line $g$ from the two points

In Desmos, enter the two points (0,3) and (4,27), then enter g(x)=6x+3 (or use the two-point slope calculation to confirm the equation).

Define a general linear $f$

Enter f(x)=m x + b (Desmos will treat $m$ and $b$ as sliders).

Build the adjusted function from $f$

Enter G(x)=2f(x+3)-f(x-1).

Solve for $m$ and $b$ using the given points on $g$

Enter the equations G(0)=3 and G(4)=27. Desmos will display the values of $m$ and $b$ that satisfy both, then choose the matching equation for $f$ from the answer choices.

Step-by-step Explanation

Model $f$ as a linear function and simplify $g$

Since $f$ is linear, let

f(x)=mx+b.

Then

\begin{aligned} g(x)&=2f(x+3)-f(x-1)\\ &=2\big(m(x+3)+b\big)-\big(m(x-1)+b\big)\\ &=2(mx+3m+b)-(mx-m+b)\\ &=(2mx-mx)+(6m+m)+(2b-b)\\ &=mx+(7m+b). \end{aligned}

So the slope of $g$ is $m$ , and the $y$ -intercept of $g$ is $7m+b$ .

Find the equation of $g$ from the given points

The line $g$ passes through $(0,3)$ and $(4,27)$ , so its slope is

\frac{27-3}{4-0}=\frac{24}{4}=6.

Since $g(0)=3$ , the $y$ -intercept is $3$ , so

g(x)=6x+3.

Match the slope to find $m$

From Step 1, $g(x)=mx+(7m+b)$ has slope $m$ . From Step 2, $g(x)=6x+3$ has slope $6$ .

m=6.

Match the intercept to find $b$

From Step 1, the $y$ -intercept of $g$ is $7m+b$ . From Step 2, the $y$ -intercept of $g$ is $3$ .

7m+b=3.

Substitute $m=6$ :

7(6)+b=3\quad\Rightarrow\quad 42+b=3\quad\Rightarrow\quad b=-39.

Therefore,

f(x)=6x-39.

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