Question

15
$f(x) = 5(3)^{x+3}$
for the given function $f$, which of the following equivalent forms shows the $y$-coordinate of the $y$-intercept of the graph of $y = f(x)$ in the $xy$-plane as a constant or coefficient?
a) $f(x) = \frac{5}{3}(3)^{x + 4}$
b) $f(x) = 15(3)^{x + 2}$
c) $f(x) = 45(3)^{x + 1}$
d) $f(x) = 135(3)^{x}$

Explanation:

Step1: Recall the y - intercept definition

The y - intercept of a function \(y = f(x)\) occurs when \(x = 0\). So we need to find the value of \(f(0)\) for the original function and then check which of the options has the y - intercept (the value when \(x = 0\)) as a constant or coefficient.

First, find \(f(0)\) for the original function \(f(x)=5(3)^{x + 3}\). Substitute \(x = 0\) into the function:
\(f(0)=5(3)^{0 + 3}=5\times3^{3}\)
Calculate \(3^{3}=27\), then \(f(0)=5\times27 = 135\).

Step2: Analyze each option

• Option A: \(f(x)=\frac{5}{3}(3)^{x + 4}\). Substitute \(x = 0\):

\(f(0)=\frac{5}{3}(3)^{0+4}=\frac{5}{3}\times3^{4}=\frac{5}{3}\times81 = 5\times27=135\)? Wait, no, \(3^{4}=81\), \(\frac{5}{3}\times81 = 135\)? Wait, no, \(\frac{5}{3}\times81=5\times27 = 135\)? Wait, but let's check the form. Wait, the form of the function is \(f(x)=a(3)^{x + b}\). We need the y - intercept (when \(x = 0\)) to be the coefficient \(a\) (if the exponent is \(x\)) or related. Wait, maybe a better approach is to rewrite the original function using exponent rules.

Recall the exponent rule \(a^{m + n}=a^{m}\times a^{n}\). So \(3^{x+3}=3^{x}\times3^{3}=27\times3^{x}\). Then the original function \(f(x)=5\times27\times3^{x}=135\times3^{x}\).

Now let's check each option:

• Option A: \(f(x)=\frac{5}{3}(3)^{x + 4}=\frac{5}{3}\times3^{x}\times3^{4}=\frac{5}{3}\times81\times3^{x}=135\times3^{x}\)? Wait, \(\frac{5}{3}\times81 = 135\), but the exponent is \(x + 4\), when \(x = 0\), \(f(0)=\frac{5}{3}\times3^{4}=\frac{5}{3}\times81 = 135\), but the form is \(\frac{5}{3}(3)^{x + 4}\), the coefficient is \(\frac{5}{3}\), not 135.
• Option B: \(f(x)=15(3)^{x + 2}=15\times3^{x}\times3^{2}=15\times9\times3^{x}=135\times3^{x}\)? Wait, \(15\times9 = 135\), but the exponent is \(x + 2\), when \(x = 0\), \(f(0)=15\times3^{2}=15\times9 = 135\), but the coefficient is 15, not 135.
• Option C: \(f(x)=45(3)^{x + 1}=45\times3^{x}\times3^{1}=45\times3\times3^{x}=135\times3^{x}\)? \(45\times3 = 135\), when \(x = 0\), \(f(0)=45\times3^{1}=135\), but the coefficient is 45, not 135.
• Option D: \(f(x)=135(3)^{x}\). When \(x = 0\), \(f(0)=135\times3^{0}=135\times1 = 135\), and the coefficient is 135, which is the y - intercept (since when \(x = 0\), \(y = 135\)). Also, we can rewrite the original function as follows:

Original function: \(f(x)=5(3)^{x + 3}\)

Using \(a^{m + n}=a^{m}\times a^{n}\), \(3^{x+3}=3^{x}\times3^{3}=27\times3^{x}\)

So \(f(x)=5\times27\times3^{x}=135\times3^{x}\), which is option D.

Answer:

D. \(f(x) = 135(3)^{x}\)