Question 180·200 Super-Hard SAT Math Questions·Advanced Math

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$x$	$g(x)$
1	4
5	-1

The table gives two values of $x$ and their corresponding values of $g(x)$ , where

g(x)=\frac{f(x)+2x-5}{x-3}

and $f$ is a linear function. Which choice is the $y$ -coordinate of the $y$ -intercept of the graph $y=f(x)$ in the $xy$ -plane?

A linear function can be hidden inside a fraction and still be determined by two input-output facts. Here the table gives outputs of $g$ , so substitute each row into the fraction before fitting $f(x)=ax+b$ . An element-list regression in Desmos finds $a$ and $b$ together. Treating the table’s outputs as values of $f$ fits the wrong line.

Hints

Hint 1
Linear means $f(x)$ has the form $ax+b$ . At $x=0$ , the $ax$ term is zero, so the requested coordinate is $b$ . How can the two table rows help you find it?
Hint 2
Each row gives an output of $g$ , not $f$ . Put the row’s $x$ and $g(x)$ into the given fraction. What equation does each row give about $f$ ?
Hint 3
An element-list regression can make both row equations true at once. Match a list of the two fraction outputs to the list of table outputs using $\sim$ . Which fitted constant is $f(0)$ ?

Step-by-step

Fit the hidden line in Desmos
Step 1
Identify the intercept you need
Because $f$ is linear, write $f(x)=ax+b$ , where $a$ and $b$ are constants. A y-intercept is where $x=0$ . So $f(0)=b$ : the target is $b$ , not an output from the $g$ table.
Step 2
Turn the table rows into equations about $f$
The rows say $g(1)=4$ and $g(5)=-1$ . Substitute each input and output into the given fraction:
$\frac{f(1)+2(1)-5}{1-3}=4,\qquad \frac{f(5)+2(5)-5}{5-3}=-1.$
The table gives values of $g$ , not values of $f$ . Using $(1,4)$ and $(5,-1)$ as points on $f$ would fit the wrong line.
Step 3
Fit both equations and read the intercept
Type $f(x)=ax+b$ , then $\left[\frac{f(1)+2(1)-5}{1-3},\frac{f(5)+2(5)-5}{5-3}\right]\sim[4,-1]$ . The regression symbol $\sim$ tells Desmos to fit $a$ and $b$ so both rows work. Under PARAMETERS, it shows $a=-0.5$ and $b=-4.5$ . Type $b$ on the next line and tap its fraction button to see $-\frac{9}{2}$ . So the $y$ -coordinate of $f$ ’s $y$ -intercept is $-\frac{9}{2}$ . Choice D.

Question 180·200 Super-Hard SAT Math Questions·Advanced Math

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