Question 82·200 Super-Hard SAT Math Questions·Advanced Math

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$x$	$f(x)$
$-4$	$-3$
$2$	$0$
$6$	$7$

Three points on the quadratic function $f(x)$ are given in the table. If $g(x)=f(x-3)$ , which choice is the $y$ -coordinate of the $y$ -intercept of the function $g(x)$ ?

Three points on a quadratic can determine its rule through quadratic regression. If a second function shifts that rule, first find which input of the original function gives the new graph’s $y$ -intercept. Then fit the quadratic in Desmos and evaluate it at that input. The inside sign changes the input, so reversing it evaluates the wrong place.

Hints

Hint 1
The $y$ -intercept is where a graph meets the vertical axis, so its input is $0$ . Put $0$ into $g(x)$ : what input does the expression inside $f$ become?
Hint 2
A quadratic regression finds the three coefficients in $ax^2+bx+c$ from three paired points. Keep each table row together when you enter the inputs in $x_1$ and outputs in $y_1$ .
Hint 3
After the regression, define $f(x)$ using the stored letters $a$ , $b$ , and $c$ . Then evaluate it at the input you found, rather than copying rounded coefficients from the screen.

Step-by-step

Fit the quadratic and evaluate the shifted input
Step 1
Find the original input you need
The $y$ -intercept is where the graph of $g$ meets the vertical axis, so its input is $0$ . Put $0$ into the new function, then use the entire expression inside $f$ as its input: $g(0)=f(0-3)=f(-3)$ . The minus sign stays; $f(3)$ uses the opposite input.
Step 2
Enter the three paired points
In Desmos, make a table with the given $x$ -values in $x_1$ and their $f(x)$ -values in $y_1$ . Desmos shows the rows $(-4,-3)$ , $(2,0)$ , and $(6,7)$ . Keep each input beside its output so the fit uses the points the question gives.
Step 3
Fit the quadratic
Type $y_1\sim ax_1^2+bx_1+c$ . The $\sim$ tells Desmos to find the coefficients, the fixed numbers in the quadratic rule, that fit the table. Under PARAMETERS, Desmos reports $a=0.125$ , $b=0.75$ , and $c=-2$ . All three points determine this fit.
Step 4
Define the fitted function
Type $f(x)=ax^2+bx+c$ on the next line. Desmos now recognizes $f$ as the quadratic from the table. Using the stored coefficients avoids retyping rounded decimals before the evaluation.
Step 5
Evaluate the intercept’s output
Type $f(-3)$ . Desmos prints $-3.125$ ; use its fraction button to see $-\frac{25}{8}$ . Since $g(0)=f(-3)$ , the $y$ -coordinate of $g$ ’s $y$ -intercept is $-\frac{25}{8}$ . Choice A.

Question 82·200 Super-Hard SAT Math Questions·Advanced Math

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