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.

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.

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

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.

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.

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

Substitute $x=0$ into the given function:

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

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

Then

Compute $6^4$:

So $s = 1296$.

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:

So

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

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

From Step 2, $s = 1296$. Substitute this into the factored form from Step 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) $y = 1296\,\cdot 216^{-\,2x/3}$.