Question 550·Hard·Math

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The function $p$ is defined by

p(x)=k\cdot 3^{mx}

for constants $k$ and $m$ . If $p(2)=12$ and $p(5)=324$ , what is $p(8)$ ?

In an exponential function, equal changes in the input multiply the output by the same factor. If the model has unknown constants, you can fit them from two given outputs with a Desmos regression, then evaluate the function. You can also compare the input gaps and reuse a multiplier. The trap is applying a one-unit factor to a longer jump.

Hints

Hint 1
The two given outputs belong to the same exponential function. Substitute each input into the formula to make two equations involving $k$ and $m$ .
Hint 2
A regression can find constants that satisfy several conditions together. In Desmos, put $p(2)$ and $p(5)$ in a list, then match them to their outputs in the same order using $\sim$ .
Hint 3
Function evaluation means putting the requested input into the fitted function. Once Desmos has found $k$ and $m$ , what could you type to get the output at $x=8$ ?

Step-by-step

Approach 1: Fit the constants in Desmos
Step 1
Turn the two outputs into equations
Putting each given input into $p(x)=k\cdot3^{mx}$ gives two conditions on the same constants:
$p(2)=k\cdot3^{2m}=12,$ $p(5)=k\cdot3^{5m}=324.$
Step 2
Fit both constants at once
Type $p(x)=k\cdot3^{mx}$ , then $[p(2),p(5)]\sim[12,324]$ . The symbol $\sim$ tells Desmos to fit the two unknown constants so both conditions hold. Under PARAMETERS, Desmos shows $k\approx1.3333$ and $m=1$ . Keep its full fitted values rather than rounding $k$ yourself.
Step 3
Evaluate the requested input
Type $p(8)$ beneath the regression. Desmos shows $8\,748$ , using the fitted constants without rounding them. So the value of $p(8)$ is $8\,748$ . Choice B.

Approach 2: Reuse the three-unit multiplier
Step 1
Compare the input gaps
The inputs go from $2$ to $5$ , then from $5$ to $8$ : each gap is $3$ . In $k\cdot3^{mx}$ , raising $x$ by $3$ adds $3m$ to the exponent, so it multiplies the output by $3^{3m}$ each time. Equal input gaps in an exponential function have the same multiplier.
Step 2
Find that multiplier
To find the factor from $p(2)$ to $p(5)$ , type $324/12$ . Desmos shows $27$ , so each three-unit jump multiplies the output by $27$ .
Step 3
Apply it to the next jump
The jump from $5$ to $8$ is also three units. Type $324(324/12)$ ; Desmos shows $8\,748$ . Multiplying $324$ by $3$ only once would stop after one unit, at $x=6$ . So $p(8)=8\,748$ . Choice B.

Question 550·Hard·Math

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