The linear function passes through the points and . A new function is defined by Which equation defines ?

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SAT Linear Functions Question 66 | Free SAT Prep | Aniko

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Question 66·Hard·Linear Functions

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The linear function $g$ passes through the points $(2,-3)$ and $(8,15)$ .

A new function is defined by

r(x)=g(3x-4)+2.

Which equation defines $r$ ?

For problems where a new function is defined from an old one, first write a clean formula for the original function (here, find the slope and intercept from two points). Then carefully apply the transformation: replace every $x$ in the original formula with the new input (such as $3x - 4$ ), perform any additional operations (like adding 2), and simplify. Pay particular attention to distributing and combining constant terms, since that is where small sign errors often create wrong-choice traps.

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Start with the original line g

Use the two points $(2,-3)$ and $(8,15)$ to find the slope of the linear function $g$ .

Find g(x) in slope-intercept form

Once you know the slope, write $g(x) = mx + b$ and plug in one of the points to solve for $b$ .

Apply the transformation to define r(x)

The new function is $r(x) = g(3x - 4) + 2$ . Think about what you do to the formula for $g(x)$ when the input $x$ is replaced by $3x - 4$ , and then what adding 2 does.

Simplify carefully

After substituting $3x - 4$ into $g$ , distribute and combine like terms step by step, watching the constant terms closely.

Desmos Guide

Use Desmos to find g(x)

In one expression line, type (15 - (-3))/(8 - 2) to confirm the slope is 3. Then type g(x) = 3x - 9 as the equation of the line through the two points.

Define r(x) in Desmos

In a new line, enter r(x) = g(3x - 4) + 2. Desmos will display an expression for $r(x)$ when you click on it or hover over it in the expressions list.

Compare with the answer choices

Either mentally expand the expression Desmos shows for $r(x)$ or type expand(g(3x - 4) + 2) in another line to see the simplified linear form. Match this simplified line with the option that has the same slope and y-intercept.

Step-by-step Explanation

Find the slope of g

The function $g$ is linear and passes through $(2,-3)$ and $(8,15)$ .

Compute the slope $m$ using

m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{15 - (-3)}{8 - 2}.

m = \frac{18}{6} = 3.

Write the equation for g(x)

Use slope-intercept form $g(x) = mx + b$ with slope $m = 3$ :

g(x) = 3x + b.

Plug in one of the points, for example $(2,-3)$ :

-3 = 3(2) + b \\[0.7em] -3 = 6 + b \\[0.7em] b = -9.

g(x) = 3x - 9.

Substitute into the definition of r(x)

We are given

r(x) = g(3x - 4) + 2.

Replace $g(x)$ with the formula $g(x) = 3x - 9$ , but use $3x - 4$ in place of $x$ :

g(3x - 4) = 3(3x - 4) - 9.

r(x) = 3(3x - 4) - 9 + 2.

Simplify to get the final equation for r(x)

Distribute the 3 and combine like terms:

3(3x - 4) - 9 + 2 = 9x - 12 - 9 + 2.

Combine the constant terms:

-12 - 9 + 2 = -19.

r(x) = 9x - 19,

which corresponds to choice A.

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation