SAT Nonlinear Functions Question 123 | Free SAT Prep

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Question 123·Hard·Nonlinear Functions

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The function $f$ is defined by

f(x)=a(x+2)(x-8),

where $a$ is a nonzero constant. What is the $x$ -coordinate of the vertex of the graph of $f$ ?

Your answer

When a quadratic is written as a nonzero constant times two linear factors, first read the $x$ -intercepts directly from the factors. The vertex lies on the axis of symmetry, so its $x$ -coordinate is the average of the intercepts' $x$ -coordinates. This avoids expanding the expression or using a more complicated vertex formula.

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Use the factors

Set each factor of $a(x+2)(x-8)$ equal to $0$ to identify the two $x$ -intercepts.

Use parabola symmetry

The vertex of a parabola is centered horizontally between its two $x$ -intercepts. Find the average of their $x$ -coordinates.

Desmos Guide

Graph a representative parabola

Because multiplying by a nonzero value of $a$ does not change the vertex's $x$ -coordinate, enter $y=(x+2)(x-8)$ into Desmos.

Locate the vertex

Click or tap the lowest point of the graph to display the vertex. Read its $x$ -coordinate to verify the result.

Step-by-step Explanation

Identify the x-intercepts

Because $a$ is nonzero, the zeros of $f$ occur when either factor is zero:

x+2=0 \quad\text{or}\quad x-8=0.

Thus, the $x$ -intercepts have $x$ -coordinates $-2$ and $8$ .

Find the axis of symmetry

For a parabola, the axis of symmetry lies halfway between its two $x$ -intercepts. Its $x$ -coordinate is

\frac{-2+8}{2}=3.

The vertex lies on this axis of symmetry, so its $x$ -coordinate is $3$ .

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