Question 133·Hard·Advanced Math

Because $(0,5)$ is on the graph, putting $0$ into $f$ gives $5$:

Any positive base to the zeroth power is $1$, so:

The output at zero is not $c$ alone. Answering $5$ would leave out $a$.

Put the three points into a Desmos table, with inputs in $x_1$ and outputs in $y_1$. Then type $y_1\sim ab^{x_1}+c$. The regression uses $\sim$ to find constants that fit all three pairs. Under PARAMETERS, Desmos shows $a=3$, $b=5$, and $c=2$. The added constant is $c$, so its value is $2$. Choice A.

The added $c$ appears in both outputs, so subtracting cancels it. From $x=0$ to $x=1$, write the change:

Cancel $c$ and subtract:

Factor out $a$:

Use the points at $x=0$ and $x=3$ to make another difference with the same factor $a$. Write the change:

Cancel $c$ and subtract:

Factor out $a$:

The first difference is $12$, not $0$, so $a(b-1)\ne0$ and division is allowed. Divide the two differences:

Cancel $a$ and divide the numbers:

Factor $b^3-1$:

Cancel $b-1$, which the question says is not zero:

The second difference spans $0$ to $3$ while the first spans $0$ to $1$, so $372/12$ is not itself the base.

Type $b^2+b+1\sim31\{b>0\}$ in Desmos. The braces enforce the given positive-base condition; without it, the equation also has a negative solution. Desmos shows $b=5$ under PARAMETERS.

Use the first difference, $a(b-1)=12$. Divide by $b-1$:

Add $a=\frac{12}{b-1}$ below the regression in Desmos. With $b=5$, it shows $a=3$.

The point at zero gave $5=a+c$. Subtract $a$:

Type $5-a$ on the next Desmos line. It shows $2$, so the added constant is $c=2$. Choice A.