Question 798·Medium·Math

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The graph shows the quadratic function $g$ .

Which choice could be the equation of $g(x)$ ?

A quadratic graph with readable $x$ -intercepts invites factored form: each intercept tells you one factor. A point off the $x$ -axis then determines the multiplier in front. Check the signs inside the factors by making sure each factor equals $0$ at its intercept, rather than copying the signs you see on the graph.

Hints

Hint 1
An $x$ -intercept is where the output is $0$ . A product is $0$ when one of its factors is $0$ . What factor becomes $0$ at $x=-1$ ?
Hint 2
The $y$ -intercept has input $x=0$ . Substitute $0$ into your two-factor expression and use the graph’s output there to find the number multiplying the factors.

Step-by-step

Build the equation from the intercepts
Step 1
Turn the x-intercepts into factors
The $x$ -intercepts are $(-1,0)$ and $(5,0)$ , where the output is $0$ . The factor $x+1$ becomes $0$ at $-1$ , and $x-5$ becomes $0$ at $5$ , so the equation has the form
$g(x)=a(x+1)(x-5).$
The number $a$ is a multiplier still to be found. Don’t copy the intercept signs into the factors; each factor must equal $0$ at its intercept.
Step 2
Use the y-intercept to make an equation
The $y$ -intercept $(0,-5)$ means $g(0)=-5$ : input $0$ gives output $-5$ . Put $0$ into the factored form to find the missing multiplier:
$-5=a(0+1)(0-5).$
Step 3
Find the multiplier
Type $-5\sim a(0+1)(0-5)$ , replacing $=$ with $\sim$ so Desmos finds the value of $a$ that fits the equation. Desmos reports $a=1$ under PARAMETERS. So the graph needs no extra multiplier in front of the factors.
Step 4
Choose the equation
Put $a=1$ into the factored form:
$g(x)=(x+1)(x-5).$
The intercepts give the factors; another point gives the multiplier. This equation matches the graph’s intercepts, so it could be $g(x)$ . Choice A.

Question 798·Medium·Math

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Step-by-step

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