Question 454·Hard·Math

0 / 2000

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?

The y-intercept of an exponential function is its output at $x=0$ . Evaluate the given function there in Desmos, then split its exponent so the constant power can become a coefficient in front. Watch where the number is written: a number used as the base of a power is not a coefficient, even if the equation is equivalent.

Hints

Hint 1
A y-intercept is where the graph crosses the vertical axis, so the input is $0$ . What value do you get when you evaluate the given function at $x=0$ ?
Hint 2
The exponent rule $a^{u+v}=a^u a^v$ turns one power into two factors. Split the exponent into a constant part and an $x$ -part. Which factor could become the coefficient in front?
Hint 3
A coefficient multiplies an expression; a base is raised to a power. Check whether the intercept value is in front of the power, rather than inside it.

Step-by-step

Find the intercept, then split the power
Step 1
Identify the input at the intercept
The y-intercept $(0,s)$ is on the vertical axis, so its input is $x=0$ . Call the given output $f(x)$ . Then $s=f(0)$ .
Step 2
Evaluate the intercept in Desmos
Type $f(x)=216^{(4-2x)/3}$ , then $f(0)$ . Desmos prints $1296$ , so the intercept value is $s=1296$ .
Step 3
Separate the constant and variable parts
Divide both terms in the exponent’s numerator by $3$ :
$\frac{4-2x}{3}=\frac{4}{3}-\frac{2x}{3}.$
Step 4
Move the constant power in front
Apply $a^{u+v}=a^u a^v$ to make the constant power a factor in front:
$\begin{aligned}y&=216^{\frac{4}{3}-\frac{2x}{3}}\y&=216^{4/3}\cdot216^{-2x/3}.\end{aligned}$
At $x=0$ , the variable factor is $216^0=1$ , so the factor in front is $s$ .
Step 5
Display the intercept as the coefficient
Replace $216^{4/3}$ with $s=1296$ :
$y=1296\cdot216^{-2x/3}.$
Here $1296$ is the coefficient, the number multiplying the power. Writing $1296$ as a base instead would not display it as the coefficient. Choice D.

Question 454·Hard·Math

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