Question 510·Medium·Math

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The graph shows the function $f$ .

Which choice gives an equation for $f(x)$ ?

A parabola is the graph of a quadratic function, and its $x$ -intercepts show where the output is $0$ . Use factored form to turn those intercepts into factors, then use a point off the $x$ -axis to find the outside multiplier with a Desmos regression. The zeros fix the factors; another point fixes their multiplier. Matching the intercepts alone doesn't guarantee the right height.

Hints

Hint 1
A zero is an input that makes the output $0$ . At each marked $x$ -intercept, which expression involving $x$ becomes $0$ ? Those expressions give you the factors of the equation.
Hint 2
The $y$ -intercept has input $0$ . Substitute that input into the factors and match the marked output to find the outside multiplier. Using a zero instead would give you $0=0$ , which can't determine it.

Step-by-step

Use the intercepts and one more point
Step 1
Build the factors from the zeros
The graph crosses the $x$ -axis at $(1,0)$ and $(5,0)$ , so $1$ and $5$ are zeros, or inputs that make $f(x)=0$ . A product is $0$ when one of its factors is $0$ , so
$f(x)=a(x-1)(x-5).$
These factors vanish at the marked zeros. The multiplier $a$ is still unknown.
Step 2
Turn the other marked point into an equation
The $y$ -intercept $(0,10)$ says input $0$ gives output $10$ , or $f(0)=10$ . Put that into the factored equation:
$10=a(0-1)(0-5).$
This point isn't a zero, so it gives an equation that can determine $a$ .
Step 3
Find the multiplier
Type $10\sim a(0-1)(0-5)$ in Desmos. The $\sim$ tells Desmos to find the unknown coefficient $a$ ; under PARAMETERS, it reports $a=2$ . Put that value outside the factors. The graph's equation is $f(x)=2(x-1)(x-5)$ . Choice A.

Question 510·Medium·Math

Strategy

Hints

Step-by-step

Lessons that teach this

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Strategy

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Step-by-step

Lessons that teach this