Question

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 given options has the same value of \(f(0)\) (since the y - coordinate of the y - intercept is \(f(0)\)) and also has this value 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 \(3^{3}=27\), so \(f(0)=5\times27 = 135\).

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, \(\frac{5}{3}\times81 = 5\times27 = 135\)? Wait, \(3^{4}=81\), \(\frac{5}{3}\times81=5\times27 = 135\)? Wait, but let's check the form. The function is \(\frac{5}{3}(3)^{x + 4}\), when \(x = 0\), it's \(\frac{5}{3}\times3^{4}\), but the coefficient here is \(\frac{5}{3}\), not 135.

• Option B: \(f(x)=15(3)^{x + 2}\). Substitute \(x = 0\):

\(f(0)=15(3)^{0 + 2}=15\times3^{2}=15\times9 = 135\)? No, \(15\times9 = 135\)? Wait, \(15\times9 = 135\)? Wait, \(3^{2}=9\), \(15\times9 = 135\)? Wait, no, \(15\times9=135\)? Wait, 159 is 135? Wait, 109 = 90, 5*9=45, 90 + 45=135. But the coefficient here is 15, not 135.

• Option C: \(f(x)=45(3)^{x + 1}\). Substitute \(x = 0\):

\(f(0)=45(3)^{0+1}=45\times3 = 135\)? No, \(45\times3 = 135\)? Wait, 453 = 135? Wait, 403=120, 5*3 = 15, 120+15 = 135. But the coefficient here is 45, not 135.

• Option D: \(f(x)=135(3)^{x}\). Substitute \(x = 0\):

\(f(0)=135(3)^{0}=135\times1 = 135\). And in this function, the coefficient is 135, which is the y - coordinate of the y - intercept (since when \(x = 0\), \(y = 135\times1=135\)). Also, the function is in the form \(f(x)=a(b)^{x}\), where \(a = 135\) is the coefficient, and when \(x = 0\), \(y=a\), so this form shows the y - intercept (when \(x = 0\), \(y = 135\)) as the coefficient.

Answer:

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