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Question 114·Hard·Nonlinear Functions

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The function is defined by

y \,=\, 216^{\,(4-2x)/3}

The graph of this equation in the $xy$ -plane has a $y$ -intercept of $(0,\,s)$ .
Which of the following equivalent equations displays the value of $s$ as the coefficient?

For exponential-function questions asking about y-intercepts, immediately substitute $x=0$ to find the intercept value. Then use exponent rules—rewrite exponents as sums (like constant + variable part) and apply $a^{m+n} = a^m a^n$ —to factor out the constant part as a separate multiplier. That constant multiplier should match the y-intercept and appear as the coefficient in the correct answer choice, so you can quickly scan the options for the one that is both algebraically equivalent and has that coefficient visible out front.

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Locate the y-intercept

To find the y-intercept $(0,s)$ of a graph, what value of $x$ should you substitute into the equation, and what does that tell you about $s$ for this function?

Rewrite the exponent

Try rewriting $\dfrac{4-2x}{3}$ as a sum of two terms, one that is just a constant and one that involves $x$ . Then think about how $a^{m+n}$ can be split.

Think about the coefficient

Once you have written $y$ as a product of two powers of 216, which factor represents a constant (the same for all $x$ )? That constant should be the y-intercept and should appear as the coefficient in the correct choice.

Desmos Guide

Graph the original function

Type y = 216^((4-2x)/3) into Desmos. This is the given function whose y-intercept and equivalent forms you are analyzing.

Graph each answer choice for comparison

On new lines, enter:

y = 6^(8-4x)
y = 216^((4/3)*(1-x))
y = 1296^(1 - x/2)
y = 1296*216^(-2x/3) Check which of these graphs lies exactly on top of the original graph for many $x$ -values; that one is equivalent to the original function.

Use Desmos to see the y-intercept and coefficient

Click on the original graph and look at the y-intercept point $(0, y)$ . Note the numerical value of this $y$ -coordinate. Then look at the equivalent equation(s) from Step 2 and see which one shows that same value as a number multiplied in front (the coefficient) rather than hidden inside an exponent or base.

Step-by-step Explanation

Find the y-intercept in terms of the function

The y-intercept occurs where $x=0$ .

Substitute $x=0$ into the given function:

y = 216^{(4-2x)/3} \Rightarrow y = 216^{(4-2\cdot 0)/3} = 216^{4/3}.

So the y-intercept is $(0,s)$ where

s = 216^{4/3}.

Simplify the y-intercept value s

To simplify $216^{4/3}$ , notice that $216 = 6^3$ .

Then

216^{4/3} = (6^3)^{4/3} = 6^{3\cdot (4/3)} = 6^4.

Compute $6^4$ :

6^2 = 36, \quad 6^4 = 36^2 = 1296.

So $s = 1296$ .

Rewrite the function to separate the constant part of the exponent

We want an equivalent equation that makes the y-intercept $s$ appear as a coefficient (a number multiplied out in front).

First rewrite the exponent:

\frac{4-2x}{3} = \frac{4}{3} - \frac{2x}{3}.

y = 216^{(4-2x)/3} = 216^{4/3 - 2x/3}.

Use the exponent rule $a^{m+n} = a^m\cdot a^n$ with $m = 4/3$ and $n = -2x/3$ :

y = 216^{4/3}\cdot 216^{-2x/3}.

Now you can see the structure: the function is “(a constant) times $216^{-2x/3}$ ,” and that constant should equal $s$ .

Insert the value of s and match the answer choice

From Step 2, $s = 1296$ . Substitute this into the factored form from Step 3:

y = 216^{4/3}\cdot 216^{-2x/3} = 1296\cdot 216^{-2x/3}.

This equation is equivalent to the original and clearly shows the y-intercept $s=1296$ as the coefficient. Among the answer choices, this matches

D) $y = 1296\,\cdot 216^{-\,2x/3}$ .

Question 114·Hard·Nonlinear Functions

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation