The graph of is shown. Which equation defines function ?

Solution available with step-by-step explanation

SAT Linear Equations in Two Variables Question 102 | Free SAT Prep | Aniko

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Question 102·Hard·Linear Equations in Two Variables

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The graph of $y = 2f(2x-1) - 2x + 1$ is shown. Which equation defines function $f$ ?

When a graph shows a linear expression built from an unknown function (for example, $g(x)=2f(2x-1)-2x+1$ ), first treat the graphed line as its own function $g(x)$ and find its equation from two points. Next, algebraically isolate the function term (here $f(2x-1)$ ). Finally, use a substitution for the input (set $t=2x-1$ ) and rewrite any remaining $x$ values in terms of $t$ before simplifying.

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Write the graphed line as $g(x)$

Use the two labeled points $(-2,5)$ and $(6,-1)$ to find the slope and intercept of the graphed line, then call it $g(x)$ .

Isolate $f(2x-1)$ first

From $g(x)=2f(2x-1)-2x+1$ , add $2x-1$ to both sides and then divide by $2$ to get an expression for $f(2x-1)$ .

Use a substitution for the input

Let $t=2x-1$ so the left side becomes $f(t)$ . Don’t forget that this also means $x=\frac{t+1}{2}$ wherever $x$ appears inside $g(x)$ .

Desmos Guide

Define the graphed line $g(x)$

Use the two points $(-2,5)$ and $(6,-1)$ to compute the slope and intercept, then enter $g(x)=-\tfrac34x+\tfrac72$ .

Rewrite $f$ using a substitution

Let the input to $f$ be $x$ (playing the role of $t$ ). Since $t=2u-1$ , we have $u=\frac{x+1}{2}$ , so enter

f(x)=\frac{g\left(\frac{x+1}{2}\right)+x}{2}.

Match to an answer choice

Desmos will simplify $f(x)$ to a linear expression. Compare it directly to the four choices to select the matching equation.

Step-by-step Explanation

Find the equation of the graphed line

Using points $(-2,5)$ and $(6,-1)$ , the slope is

m=\frac{-1-5}{6-(-2)}=\frac{-6}{8}=-\tfrac34.

Using $(-2,5)$ to find $b$ :

5=\left(-\tfrac34\right)(-2)+b=\tfrac32+b \implies b=\tfrac72.

So the graphed line is $g(x)=-\tfrac34x+\tfrac72$ .

Isolate the function expression

The graph shows

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

Solve for $f(2x-1)$ :

2f(2x-1)=g(x)+2x-1 \implies f(2x-1)=\frac{g(x)+2x-1}{2}.

Substitute a new variable for the input to $f$

Let $t=2x-1$ . Then $x=\frac{t+1}{2}$ and

f(t)=\frac{g\left(\frac{t+1}{2}\right)+t}{2}

because $2x-1=t$ .

Compute $g\left(\frac{t+1}{2}\right)$ and simplify

Compute

\begin{aligned} g\left(\frac{t+1}{2}\right) &=-\tfrac34\left(\frac{t+1}{2}\right)+\tfrac72 \\ &=-\tfrac38(t+1)+\tfrac72 \\ &=-\tfrac38t-\tfrac38+\tfrac{28}{8} \\ &=-\tfrac38t+\tfrac{25}{8}. \end{aligned}

Then

\begin{aligned} f(t) &=\frac{\left(-\tfrac38t+\tfrac{25}{8}\right)+t}{2} \\ &=\frac{\tfrac58t+\tfrac{25}{8}}{2}. \end{aligned}

State the final equation for $f$

f(t)=\tfrac{5}{16}t+\tfrac{25}{16}.

Renaming $t$ as $x$ , $f(x)=\tfrac{5}{16}x+\tfrac{25}{16}$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation