The function is defined by In the -plane, the graph of function is obtained by reflecting the graph of across the -axis and then translating it units to the right. Which equation defines ?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 110 | Free SAT Prep | Aniko

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Question 110·Medium·Nonlinear Functions

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The function $q$ is defined by

q(x) = (x + 2)^2

In the $xy$ -plane, the graph of function $t$ is obtained by reflecting the graph of $q$ across the $x$ -axis and then translating it $5$ units to the right.

Which equation defines $t$ ?

For transformation questions, first identify the vertex and orientation (up/down) of the given parabola in vertex form. Then apply transformations in the order given: reflections change signs (e.g., across the $x$ -axis multiplies the whole function by $-1$ ), while horizontal shifts change the input ( $x$ becomes $x-h$ for a shift right by $h$ ). Track how the vertex moves step by step, and finally match both the new vertex and the opening direction to the answer choice written in vertex form.

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Start with the basic shape

What is the vertex and opening direction of the parabola $q(x)=(x+2)^2$ ? Think about how $x+2$ affects the horizontal position.

Effect of reflecting across the x-axis

How does reflecting a graph across the $x$ -axis change the function rule? What happens to the $y$ -values (outputs) of the function?

Effect of shifting right

For a function $f(x)$ , what does the expression $f(x-5)$ do to the graph? How does it move the vertex horizontally?

Combine both transformations

First apply the reflection to $q(x)$ , then apply the horizontal shift to that result. After you have the final algebraic expression, compare it carefully to each answer option.

Desmos Guide

Graph the original function

In Desmos, type y = (x + 2)^2. Observe that the vertex is at $(-2,0)$ and the parabola opens upward.

Apply the reflection across the x-axis

Type y = -(x + 2)^2. Compare it to the original graph: the vertex stays at $(-2,0)$ , but the graph now opens downward because of the negative sign.

Compare answer choices to the described transformation

Now enter each answer choice into Desmos (one at a time or all together):

y = -(x + 2)^2 + 5
y = -(x + 7)^2
y = (x - 3)^2
y = -(x - 3)^2 Look for the graph whose parabola is the reflected one (downward-opening) and whose vertex has moved exactly 5 units to the right from $(-2,0)$ to $(3,0)$ . The matching equation is the correct $t(x)$ .

Step-by-step Explanation

Understand the original function

The function $q(x)=(x+2)^2$ is a parabola in vertex form $y=(x-h)^2$ , where the vertex is at $(-h,0)$ .

Here, $x+2$ can be written as $x-(-2)$ , so the vertex of $q$ is at $(-2,0)$ , and the parabola opens upward (because the coefficient in front of the square is positive).

Reflect across the x-axis

Reflecting a graph across the $x$ -axis changes every $y$ -value to its opposite. Algebraically, you multiply the function by $-1$ .

So reflecting $q(x)$ across the $x$ -axis gives the intermediate function

- q(x) = -(x+2)^2.

This new parabola still has vertex $(-2,0)$ but now opens downward instead of upward.

Translate 5 units to the right

To shift any function $f(x)$ to the right by 5 units, you replace $x$ with $x-5$ , giving the new function $f(x-5)$ .

Here, our intermediate function is $-(x+2)^2$ . After shifting it 5 units to the right, the new function is

-(x-5+2)^2.

This parabola now has its vertex at $(3,0)$ and still opens downward.

Simplify and match to an answer choice

Simplify the expression from the previous step:

-(x-5+2)^2 = -(x-3)^2.

So the transformed function is defined by

t(x)=-(x-3)^2,

which matches the correct answer choice.

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