Question 20·200 Super-Hard SAT Math Questions·Advanced Math

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A function $f$ is defined for all real $x$ by

f(x)=c+a\cdot b^{x}

where $a$ , $b$ , and $c$ are constants, $b>0$ , and $b\ne 1$ . The table shows three values of $f$ .

$x$	1	3	5
$f(x)$	8	20	44

Which choice is the value of $c$ ?

Equally spaced inputs and a rule of the form $c+ab^x$ signal a shifted exponential: $c$ is added to every output. Compare outputs after subtracting $c$ , not before. On a timed test, fit the given form to the table with a Desmos regression and read $c$ under PARAMETERS. Ratios of the raw outputs ignore the shift.

Hints

Hint 1
Each table column is an input-output pair: $f(1)=8$ pairs $1$ with $8$ . Keep all three pairs together when you enter them in a Desmos table; the fitted rule must match every pair.
Hint 2
The $+c$ is a vertical shift, an amount added to every output. That means the table outputs themselves need not have a constant ratio. Fit the whole given form $c+ab^x$ , not $ab^x$ .

Step-by-step

Approach 1: Fit the given form in Desmos
Step 1
Identify what gets multiplied
The inputs go from $1$ to $3$ to $5$ , two at a time, so $b^x$ gets multiplied by $b^2$ at each step. But $c$ is added afterward. It's a vertical shift, the same amount added to every output, so don't expect the raw outputs $8$ , $20$ , and $44$ to have a constant ratio.
Step 2
Fit the table and read the shift
Enter the three pairs in a Desmos table, then type $y_1\sim c+ab^{x_1}\{b>0\}$ . Here $x_1$ and $y_1$ name the table columns, and the regression finds constants that fit them. Under PARAMETERS, Desmos shows $a\approx8.485$ , $b\approx1.414$ , and $c\approx-4$ . The question asks for the added constant, so $c=-4$ . Choice B.

Approach 2: Use the equal input steps
Step 1
Subtract the shift from each output
Each table value equals $c+ab^x$ . Subtracting the same shift $c$ leaves only the exponential part:
$\begin{aligned}8-c&=ab\\[1.4em]20-c&=ab^3\\[1.4em]44-c&=ab^5.\end{aligned}$
Step 2
Link the equally spaced values
The exponents $1$ , $3$ , and $5$ are equally spaced. Multiplying $ab$ by $b^2$ gives $ab^3$ , and multiplying by $b^2$ again gives $ab^5$ . So the middle shifted value squared equals the product of the outer two:
$(ab^3)^2=(ab)(ab^5)$
Replace those three terms with the table expressions from the previous step:
$(20-c)^2=(8-c)(44-c).$
Step 3
Solve for the shift
Type $(20-c)^2\sim(8-c)(44-c)$ in Desmos. The $\sim$ asks its regression to find the unknown constant; under PARAMETERS, it shows $c=-4$ . So the shift in $f(x)=c+ab^x$ is $-4$ . Choice B.

Question 20·200 Super-Hard SAT Math Questions·Advanced Math

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