The quadratic function is defined by A new function is defined by . At which of the following -coordinates does the graph of intersect the -axis at its greater -intercept?

Solution available with step-by-step explanation

SAT Nonlinear Functions Question 152 | Free SAT Prep | Aniko

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Question 152·Medium·Nonlinear Functions

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The quadratic function $q$ is defined by

q(x) = (x - 4)(x + 2).

A new function is defined by $r(x) = q(x - 3)$ . At which of the following $x$ -coordinates does the graph of $y = r(x)$ intersect the $x$ -axis at its greater $x$ -intercept?

For transformation questions, avoid expanding quadratics unless necessary. First, find key features (like x-intercepts) of the original function from its factored form, then apply the transformation rule: replacing $x$ with $x - h$ shifts the graph right by $h$ , so you simply add $h$ to each original x-coordinate. Finally, answer exactly what the question asks—here, the greater x-intercept—not just any intercept.

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Start with the original function q(x)

To find x-intercepts, you need the x-values that make the function equal to 0. Use the factored form $q(x) = (x - 4)(x + 2)$ to find where $q(x) = 0$ .

Use the zero-product property

If $(x - 4)(x + 2) = 0$ , what must be true about each factor? Solve $x - 4 = 0$ and $x + 2 = 0$ to get the x-intercepts of $q(x)$ .

Interpret r(x) = q(x - 3) as a shift

Think about what happens to a graph when you replace $x$ with $x - 3$ inside a function. How do the x-coordinates of key points, like x-intercepts, change?

Apply the shift to the x-intercepts

Once you know the x-intercepts of $q(x)$ and that the graph is shifted horizontally, adjust each of those x-values accordingly, then pick the larger one.

Desmos Guide

Enter the original function

In Desmos, type q(x) = (x - 4)(x + 2) to graph the original quadratic and see its x-intercepts at $x = -2$ and $x = 4$ .

Enter the transformed function

On a new line, type r(x) = q(x - 3) to graph the new function. This will show the graph of $q(x)$ shifted horizontally.

Identify the x-intercepts of r(x)

Click on the points where the graph of r(x) crosses the x-axis. Note both x-values and then determine which of these is the larger x-coordinate; that is the answer.

Step-by-step Explanation

Find the x-intercepts of q(x)

To find where the graph of $y = q(x)$ crosses the x-axis, set $q(x) = 0$ :

q(x) = (x - 4)(x + 2) = 0.

By the zero-product property, either $x - 4 = 0$ or $x + 2 = 0$ , which gives the x-intercepts of $q(x)$ as $x = 4$ and $x = -2$ .

Understand the transformation r(x) = q(x - 3)

The function $r(x) = q(x - 3)$ is formed by replacing $x$ with $x - 3$ inside $q(x)$ . This transformation shifts the entire graph of $q(x)$ horizontally to the right by 3 units.

That means every x-coordinate on the graph of $q(x)$ , including its x-intercepts at $-2$ and $4$ , will increase by 3 to give the corresponding x-coordinates on the graph of $r(x)$ .

Find the new x-intercepts and choose the greater one

Since the graph shifts 3 units to the right, add 3 to each original x-intercept:

From $x = -2$ to $x = -2 + 3 = 1$ .
From $x = 4$ to $x = 4 + 3 = 7$ .

So the x-intercepts of $y = r(x)$ are $x = 1$ and $x = 7$ , and the greater x-intercept is $x = 7$ , which corresponds to choice D.

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