Question 28·SAT Desmos Regression Practice·Advanced Math

A function value gives the output for an input. Substitute $0$ for $x$ because $p(0)=5$:

Use $4^0=1$:

So $5$ includes both $a$ and $b$, not $a$ alone.

Substitute $2$ for $x$ because $p(2)=37$:

Use $4^2=16$:

Only the power changes with the input; the added $b$ stays unchanged.

Type $[a+b,16a+b]\sim[5,37]$. A matched-list regression pairs each equation's left side with its output, so Desmos finds constants that fit both conditions. Under PARAMETERS, it reports $a\approx2.1333$ and $b\approx2.8667$.

Type $p(x)=a(4^x)+b$, then $p(1)$. Desmos prints $11.4$; tap the fraction button to see $\frac{57}{5}$. That's the output at input $1$, not either constant. Choice D.

Subtract the equations for the two given outputs so the constant term $b$ cancels:

Simplify:

So the total change in output is $32$.

From input $0$ to input $1$, the power goes from $1$ to $4$. Write the change:

Cancel $b$:

Because $3a$ is one-fifth of $15a$, use the previous change:

Add that change to $p(0)=5$:

Combine the fractions:

So the output at input $1$ is $\frac{57}{5}$. Choice D.