Question 27·200 Super-Hard SAT Math Questions·Advanced Math

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f(x)=a^x-b

The exponential function given above passes through the points $(c,20)$ and $(3c,1340)$ , where $a>0$ and $a\ne 1$ . Which choice is a possible value of $b$ ?

In a shifted exponential function, inputs where one is three times the other point to a power rule: the exponential part at the larger input is the cube of the part at the smaller input. Cube the powered part, not the entire output. Name that part, subtract the two outputs to remove the shared shift, and use a Desmos regression to find its value.

Hints

Hint 1
A point $(r,s)$ on a function's graph means $f(r)=s$ . Use each point's first coordinate as the input to write an equation for its output.
Hint 2
The power rule gives $a^{3c}=(a^c)^3$ . The $-b$ stays outside the power, so it isn't cubed along with $a^c$ .
Hint 3
Both point equations contain the same $-b$ . Subtract one equation from the other to remove it, then solve for the positive value of $a^c$ .

Step-by-step

Cube the exponential part
Step 1
Turn the points into equations
A point $(x,y)$ on the graph means the input $x$ gives the output $y$ . Substitute each given input into $f(x)=a^x-b$ :
$a^c-b=20$ $a^{3c}-b=1340.$
Step 2
Name the part that gets cubed
Let $u=a^c$ . Tripling the exponent cubes the power because $a^{3c}=(a^c)^3=u^3$ . The $-b$ is outside that power, so the equations become:
$u-b=20$ $u^3-b=1340.$
Step 3
Remove the shared shift
Subtract the first equation from the second, so the two copies of $-b$ cancel:
$(u^3-b)-(u-b)=1340-20$ $u^3-u=1320.$
Step 4
Find the positive exponential value
Since $a>0$ , its power $u=a^c$ must be positive. Type $x_1^3-x_1\sim1320\{x_1>0\}$ in Desmos, using $x_1$ for $u$ . The $\sim$ tells Desmos to fit that unknown; under PARAMETERS, it shows $x_1=11$ . So $u=11$ , which is possible with $a=11$ and $c=1$ .
Step 5
Recover the requested shift
The problem asks for $b$ , not $u$ . From $u-b=20$ , add $b$ to both sides:
$u=20+b.$
Subtract $20$ :
$b=u-20.$
Add $x_1-20$ below the regression in Desmos. It displays $-9$ , so a possible value of $b$ is $-9$ . Choice D.

Question 27·200 Super-Hard SAT Math Questions·Advanced Math

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