Question of the Day

The function $p$ is defined by $p(x) = nx^3 + 11$, where $n$ is a constant. In the $xy$-plane, the graph of $y = p(x)$ passes through the point $(-3, 65)$.

What is the value of $p(4)$?

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We are currently calling this question the "unknown value $Q$." As is often the case, we are not seeking to be creative here. So, one route to the correct answer is to begin by determining the unknown value of $n$.

Given the point $(-3, 65)$, we can plug in $-3$ for $x$ within the original function. The $65$ we swap in for $p(x)$, since $p(x)$ is $y$. Doing so, we arrive at:

$$65 = n(-3)^3 + 11$$

Subtracting $11$ from both sides, we come to:

$$54 = n(-3)^3$$

That expression can become $54 = n(-27)$. And then, when we divide both sides by $-27$, we arrive at $n = -2$.

Knowing that $n = -2$, we can rewrite the original function as:

$$p(x) = -2x^3 + 11$$

To finish things out and determine the value of $p(4)$, we execute the classic function move of inputting $4$ in for $x$ within the function:

$$-2(4)^3 + 11$$

Firing what’s above into a calculator, we come to the correct final answer of $-117$.

If your Desmos fluency is quite high, and you arrived at the correct answer here via Desmos, we’re here to validate the legitimacy of this route. That said, given what the test writers could do within this type of question, we would encourage you to be comfortable with the traditional mathematical route we mapped out above.

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