Question 678·Medium·Math

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The function $p$ is defined by $p(x)=a\,3^{x}+b$ , where $a$ and $b$ are constants. If $p(1)=10$ and $p(3)=100$ , what is the value of $p(2)$ ?

A shifted exponential has a fixed base and two unknown constants: one multiplies the power, and one is added afterward. Two known outputs give two equations for those constants. On a timed test, fit both facts in one Desmos regression, then evaluate the requested input. Multiplying an entire output by the base fails because the added constant does not multiply.

Hints

Hint 1
In $p(1)=10$ , the input is $1$ and the output is $10$ . Replace $x$ with $1$ in the rule, then do the same with $p(3)=100$ . What two equations do you get?
Hint 2
A regression can find unknown constants from known facts. Pair each output with its expression in one pair of lists, so Desmos finds the same $a$ and $b$ for both.
Hint 3
To find $p(2)$ , replace $x$ with $2$ after finding $a$ and $b$ . Keep $+b$ outside the power: changing the input multiplies the exponential part, not the added constant.

Step-by-step

Fit both outputs in Desmos
Step 1
Turn the outputs into equations
The input in $p(1)=10$ is $1$ , so replace $x$ with $1$ in the rule. Do the same for $p(3)=100$ :
$\begin{aligned}p(1)=a3^1+b=3a+b&=10,\p(3)=a3^3+b=27a+b&=100.\end{aligned}$
Both equations must use the same $a$ and $b$ .
Step 2
Find the constants together
Type $[10,100]\sim[3a+b,27a+b]$ in Desmos. The regression symbol $\sim$ tells Desmos to match each output on the left with its expression on the right, using one shared pair of constants. Under PARAMETERS, it shows $a=3.75$ and $b=-1.25$ . Fitting the equations separately would not require the constants to match.
Step 3
Evaluate the requested input
$p(2)$ means put $2$ in for $x$ . Add $a3^2+b$ below the regression; Desmos uses the fitted constants and prints $32.5$ . The $+b$ stays outside the power, so the value of $p(2)$ is $32.5$ . Choice B.

Question 678·Medium·Math

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