The graph of is shifted right by 3 units and up by 2 units to produce the graph of . Which of the following could be ?

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SAT Nonlinear Functions Question 172 | Free SAT Prep | Aniko

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Question 172·Easy·Nonlinear Functions

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The graph of $y=|x|$ is shifted right by 3 units and up by 2 units to produce the graph of $y=g(x)$ . Which of the following could be $g(x)$ ?

For absolute value and other transformation questions, focus on how the algebraic form changes the graph: expressions like $x-h$ inside the absolute value (or parentheses) shift the graph right by $h$ , $x+h$ shifts left by $h$ , and adding or subtracting a number outside the absolute value moves the graph up or down. Quickly translate the described shifts into a function form (first handle the horizontal shift inside, then the vertical shift outside), and then match that form to the answer choices without graphing point by point.

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Think about inside vs. outside the absolute value

Ask yourself: which part of the equation controls left/right movement, and which part controls up/down movement?

Right vs. left shift

For a function like $y=|x-h|$ , does a positive $h$ move the graph left or right? Use that to decide what the inside of the absolute value should look like for a shift 3 units to the right.

Up vs. down shift

For a function like $y=|x|+k$ , does a positive $k$ move the graph up or down? Use that to decide what number should be added or subtracted outside the absolute value for a shift of 2 units up.

Combine both shifts

First figure out the correct horizontal shift of $y=|x|$ , then apply the vertical shift to that new function. Look for the choice that matches both changes.

Desmos Guide

Graph the original function

In Desmos, type y = abs(x) to see the base graph of $y=|x|$ . This is your reference V-shaped graph with its vertex at $(0,0)$ .

Graph each answer choice

On separate lines, enter each option: y = abs(x+3)+2, y = abs(x-2)+3, y = abs(x-3)+2, and y = abs(x-3)-2. Each will appear as a V-shaped graph similar to $y=|x|$ but shifted.

Compare vertices and shifts

Look at where the vertex (the point of the V) of each option is located compared to the vertex of $y=|x|$ at $(0,0)$ . Identify which graph’s vertex is exactly 3 units to the right and 2 units up from $(0,0)$ ; that equation corresponds to $g(x)$ .

Step-by-step Explanation

Understand how horizontal shifts work for $y=|x|$

A horizontal shift changes the expression inside the absolute value. For a function like $y=|x-h|$ :

If $h>0$ , the graph of $y=|x|$ shifts right by $h$ units.
If $h<0$ , the graph shifts left by $|h|$ units. So, to move $y=|x|$ right by 3 units, we replace $x$ with $x-3$ to get $y=|x-3|$ .

Understand how vertical shifts work

A vertical shift changes the expression outside the absolute value. For a function like $y=|x|+k$ :

If $k>0$ , the graph shifts up by $k$ units.
If $k<0$ , the graph shifts down by $|k|$ units. So, to move a graph up by 2 units, we add 2 outside the absolute value, for example $y=|x|+2$ or, after a horizontal shift, $y=|x-3|+2$ .

Combine the right shift and the up shift

The problem says the graph of $y=|x|$ is shifted right by 3 and up by 2. Using the rules:

Right 3 units: inside becomes $x-3$ , giving $y=|x-3|$ .
Up 2 units: add $+2$ outside that result. Now compare this combined transformation to the answer choices to see which one matches exactly.

Match with the correct answer choice

From the choices, the only function that has both a right shift of 3 (inside as $x-3$ ) and an upward shift of 2 (outside as $+2$ ) is:

y=|x-3|+2

So the correct answer is $|x-3|+2$ .

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Desmos Guide

Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation