Question 490·Hard·Math

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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

A quadratic in factored form shows its zeros: the inputs that make its output $0$ . The parabola mirrors across a vertical line halfway between those zeros, so its vertex lies at their midpoint. Solve each factor for its zero, then use Desmos to average them. Don't copy the signs inside the factors as the zeros.

Hints

Hint 1
A zero is an input that makes the function equal $0$ . Because the multiplier $a$ isn't zero, either factor can make the product $0$ . What input makes $x+2$ equal $0$ ?
Hint 2
The vertex is a parabola's turning point. Its two zeros are equally far from the vertex's vertical line, so the vertex's $x$ -coordinate is their midpoint. What are the two zeros?

Step-by-step

Find the midpoint of the zeros
Step 1
Set the first factor to zero
A factor is one part of a product, and a zero is an input that makes $f(x)=0$ . Because $a$ is nonzero, the first factor gives a zero when it equals $0$ . Set it equal to $0$ :
$x+2=0$
Step 2
Solve for the first zero
Subtract $2$ :
$x=-2$
Step 3
Set the second factor to zero
The other factor also makes the product $0$ when it equals $0$ . Set it equal to $0$ :
$x-8=0$
Step 4
Solve for the second zero
Add $8$ :
$x=8$
Step 5
Locate the vertex halfway between the zeros
A parabola mirrors across a vertical line through its vertex, the turning point. So the vertex's $x$ -coordinate is halfway between the zeros, regardless of the value of $a$ .
Step 6
Calculate the vertex's x-coordinate
Type $\frac{-2+8}{2}$ in Desmos; it prints $3$ . That's the $x$ -coordinate, not an output of $f$ . Grid in 3.

Question 490·Hard·Math

Strategy

Hints

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