Question 366·Medium·Math

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

The function can be written in the form $f(x)=a(x+1)(x-5)$ , where $a$ is a constant.

Which choice is the value of $a$ ?

A factored quadratic shows where its graph meets the $x$ -axis, but those points don't reveal the multiplier in front. If the graph labels another plotted point, substitute its input and output into the given form, then use a one-unknown Desmos regression to find the multiplier. Using an $x$ -intercept instead gives $0=0$ for every multiplier.

Hints

Hint 1
In factored form, each $x$ -intercept makes one factor $0$ . Substituting either intercept would give $0=0$ , which can't tell you the value of $a$ . Which labeled point has a nonzero output?
Hint 2
A point $(u,v)$ on $y=f(x)$ means $f(u)=v$ : the first coordinate is the input, and the second is the output. For the labeled point on the vertical axis, what do you substitute for $x$ ?

Step-by-step

Use the labeled point to find the multiplier
Step 1
Choose a point that can determine the multiplier
The graph labels $(0,-10)$ at the y-intercept, where $x=0$ . Use a point not already built into the factors to find the multiplier. At either root, $-1$ or $5$ , one factor is $0$ , so the equation would read $0=0$ for every $a$ .
Step 2
Turn the point into an equation
The point $(0,-10)$ means the input is $0$ and the output is $-10$ , so $f(0)=-10$ . Put both coordinates into the given function:
$-10=a(0+1)(0-5).$
Step 3
Find the multiplier in Desmos
Type $-10\sim a(0+1)(0-5)$ . The tilde tells Desmos to find the unknown constant that makes the sides equal. Under PARAMETERS, Desmos shows $a=2$ . So the value of $a$ for the graphed function is $2$ . Choice A.

Question 366·Medium·Math

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