Question 177·200 Super-Hard SAT Math Questions·Advanced Math

At the $y$-intercept, the input to $g$ is $0$. Substitute it into the given rule:

Simplify the input inside $f$:

Zero goes into $g$; its inside expression determines the input to $f$. Putting $0$ into both functions would miss that change.

The axis of symmetry is the line through the parabola’s turning point. Since it is $x=4$, equal distances to either side of $4$ must give equal outputs. Write

Here $k$ is the output at $x=4$, and $a$ controls how the outputs change with squared distance from $4$. Both still need to be found.

A point $(u,v)$ on $y=f(x)$ means input $u$ gives output $v$. So the two given points say

These are two conditions for the two unknowns, $a$ and $k$.

Type $f(x)=a(x-4)^2+k$, then $[f(2),f(7)]\sim[5,-4]$. The regression symbol $\sim$ asks Desmos to fit the two given outputs. Under PARAMETERS, it shows $a=-1.8$ and $k=12.2$.

Type $f(-1)$ below the regression. Desmos shows $-32.8$, or $-\frac{164}{5}$ using its fraction button. This is the output to use in $g(0)$; the input is $-1$, not $0$.

Type $-2f(-1)+3$. Desmos shows $68.6$; its fraction button gives $\frac{343}{5}$. The $-2$ multiplies the output of $f$ before you add $3$. Since this expression is $g(0)$, the $y$-intercept is $(0,\frac{343}{5})$, so $b=\frac{343}{5}$. Choice A.