Question 107·200 Super-Hard SAT Math Questions·Algebra

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Let

f(x)=mx+b

and define $g(x)=f(x)+4$ .

The point $(2,10)$ lies on the graph of $y=f(x)-4$ , and the point $(6,-2)$ lies on the graph of $y=f(x-3)+1$ .

Which choice gives the equation of $g(x)$ ?

With function transformations, points on shifted graphs don't immediately give points on the original line. Undo the change inside $f$ to recover its input, and undo the change outside to recover its output. Fit the original line using a Desmos list regression, then apply the shift defining the requested line. Inside changes the input; outside changes the output.

Hints

Hint 1
A point $(a,c)$ on $y=f(x)-4$ says $f(a)-4=c$ , not $f(a)=c$ . Undo the outside subtraction to find the first output of $f$ .
Hint 2
In $f(x-3)$ , the input to $f$ is $x-3$ . At the second point's $x$ -coordinate, find that input first. Then undo the outside $+1$ to find another output of $f$ .
Hint 3
Two outputs at different inputs determine one linear function. Put both into $f(x)=mx+b$ and use a Desmos list regression to find $m$ and $b$ . Which function does the question ask for after that?

Step-by-step

Recover two points on $f$ , then fit the line
Step 1
Recover the first output of $f$
At $(2,10)$ , the graph's output $10$ equals $f(2)-4$ , not $f(2)$ . Translate the point:
$10=f(2)-4.$
Add $4$ :
$f(2)=14.$
Step 2
Recover the second input and output
At $(6,-2)$ on $y=f(x-3)+1$ , $6$ is not the input to $f$ . The input is everything inside $f(\ )$ , and the outside $+1$ changes its output. Translate the point:
$-2=f(6-3)+1.$
Evaluate the input:
$-2=f(3)+1.$
Subtract $1$ :
$f(3)=-3.$
Step 3
Find the original line in Desmos
In $f(x)=mx+b$ , $m$ is the slope, the output change for each increase of $1$ in input, and $b$ is the output at $x=0$ . Type $[2m+b,3m+b]\sim[14,-3]$ in Desmos. The lists pair the two recovered outputs with their inputs; $\sim$ tells Desmos to fit both equations. Under PARAMETERS, it shows $m=-17$ and $b=48$ .
Step 4
Apply the final shift to get $g$
Type $b+4$ on the next Desmos line; it prints $52$ . Since $g(x)=f(x)+4$ , the $y$ -intercept (the output at $x=0$ ) increases, but the slope stays $-17$ . Use $f(x)=mx+b$ and group the constants:
$g(x)=mx+(b+4).$
Insert the values Desmos found:
$g(x)=-17x+52.$
That's the equation of $g$ . Choice C.

Question 107·200 Super-Hard SAT Math Questions·Algebra

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