The graph of is shown. Which choice gives a value of for which the equation has exactly three real solutions?

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 graph of $y=f(x)$ is shown.

Which choice gives a value of $k$ for which the equation $f(x)=k$ has exactly three real solutions?

Translate $f(x)=k$ into a graph idea: it asks where the curve meets the horizontal line $y=k$ . For a cubic with a visible local maximum and local minimum, the line hits the curve three times only when $k$ is strictly between those two turning-point y-values; at exactly the turning-point y-values it hits only twice, and outside that range it hits once.

Hints

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Connect the equation to the graph

Think about what it means on the graph for $f(x)$ to equal a constant $k$ .

Look at the turning points

Find the y-value of the local maximum and the y-value of the local minimum shown on the curve.

Decide when there are three intersections

A horizontal line intersects this kind of cubic three times only when it is between those two y-values.

Desmos Guide

Add the given graph as an image

Open Desmos. Click the "+" button and choose Image. Upload the provided graph image so you can trace intersections directly.

Align the axes

Move and scale the image so the axes in the image line up with the Desmos axes. Use the labeled points $(-1,4)$ and $(2,-2)$ and the integer tick marks to help align the scale.

Create a horizontal line with a slider

Type $y=k$ and let Desmos create a slider for $k$ .

Test the choices by counting intersections

Set $k$ to each answer choice ( $-3$ , $-2$ , $0$ , $4$ ) and count how many times the line $y=k$ intersects the curve in the image. Choose the value of $k$ that creates exactly 3 intersections.

Step-by-step Explanation

Identify the key y-values from the graph

From the graph, the curve has a local maximum at $(-1,4)$ and a local minimum at $(2,-2)$ . So the highest turning-point y-value is $4$ and the lowest turning-point y-value is $-2$ .

Use intersections with a horizontal line

For a cubic with two turning points, a horizontal line $y=k$ will intersect the graph:

3 times if $-2<k<4$ (between the turning-point y-values),
2 times if $k=-2$ or $k=4$ (it just touches at a turning point),
1 time if $k<-2$ or $k>4$ .

Select the value of $k$ that gives three solutions

Among the choices, only $0$ satisfies $-2<0<4$ , so $f(x)=0$ has exactly three real solutions.

Therefore, the correct answer is $0$ .

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