Question 155·200 Super-Hard SAT Math Questions·Advanced Math

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The quadratic function $f$ has $x$ -intercepts at $x=-4$ and $x=2$ . In addition, $f(5)=9$ .

Let $g(x)=f(x-1)+f(1-x)$ . Which choice is the $y$ -intercept of the graph of $g$ ?

Two roots of a quadratic tell you its factors, but not its multiplier. An extra function value fixes that multiplier with a one-parameter Desmos regression. For a new function built from the quadratic, find its $y$ -intercept by putting $0$ into the new function, then evaluate the resulting inputs. Opposite inputs don't necessarily have equal outputs.

Hints

Hint 1
A $y$ -intercept is where a graph has input $x=0$ . The graph here belongs to $g$ , not $f$ , so put $0$ into $g(x)$ first. What two inputs will $f$ receive?
Hint 2
Each root is an input where a quadratic equals $0$ . The roots supply two factors. Keep the sign straight: what factor becomes $0$ when $x=-4$ ?
Hint 3
The roots leave an unknown multiplier in the factored form of $f$ . Use the additional input-output fact to find it. Then evaluate $f$ at the two inputs you found for $g(0)$ .

Step-by-step

Build $f$ , then evaluate $g(0)$
Step 1
Find the inputs needed for the intercept
A $y$ -intercept is where the graph crosses the vertical axis, so its input is $0$ . Put $0$ into $g$ , not directly into $f$ . Substitute into the given rule:
$g(0)=f(0-1)+f(1-0).$
Simplify the inputs:
$g(0)=f(-1)+f(1).$
The two outputs must be added; the opposite inputs don't make them equal.
Step 2
Build the quadratic from its intercepts
An $x$ -intercept is an input where $f(x)=0$ . The factor $x+4$ is $0$ at $-4$ , and $x-2$ is $0$ at $2$ , so
$f(x)=a(x+4)(x-2).$
The multiplier $a$ is still unknown: changing it keeps both intercepts in place.
Step 3
Use the extra output to find the multiplier
The given $f(5)=9$ fixes $a$ because $5$ isn't either root. Substitute the input and output:
$9=a(5+4)(5-2).$
Type $9\sim a(5+4)(5-2)$ . The $\sim$ tells Desmos to fit the unknown multiplier; it reports $a\approx0.333333$ under PARAMETERS. Type $a$ on the next line and use its fraction button to read $a=\frac13$ .
Step 4
Evaluate the first output
Type $f(x)=a(x+4)(x-2)$ , then $f(-1)$ . Desmos uses the fitted $a$ and prints $f(-1)=-3$ , the first output needed for $g(0)$ . Keep the negative input in parentheses so it goes into both factors.
Step 5
Add the second output and identify the intercept
Since $f(-1)=-3$ , the sum becomes
$g(0)=f(1)-3.$
Type $f(1)-3$ . Desmos gives $-\frac{14}{3}$ ; use its fraction button if it first displays a decimal. So the $y$ -intercept of $g$ is $(0,-\frac{14}{3})$ . Choice C.

Question 155·200 Super-Hard SAT Math Questions·Advanced Math

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