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 y - intercept definition

The y - intercept of a function \(y = f(x)\) occurs when \(x = 0\). So we need to find the form of the function where when \(x = 0\), the constant or coefficient is the value of \(f(0)\) (the y - coordinate of the y - intercept). Also, we can use the exponent rule \(a^{m + n}=a^{m}\times a^{n}\) to rewrite the given function.

Given \(f(x)=5(3)^{x + 3}\), using the rule \(a^{m + n}=a^{m}\times a^{n}\), we can rewrite \(3^{x+3}\) as \(3^{x}\times3^{3}\).

Step2: Simplify the given function

\(f(x)=5\times3^{3}\times3^{x}\)
Since \(3^{3}=27\), then \(5\times27 = 135\). So \(f(x)=135\times3^{x}\)

Now let's check each option:

• Option A: \(f(x)=\frac{5}{3}(3)^{x + 4}=\frac{5}{3}\times3^{x}\times3^{4}=\frac{5}{3}\times81\times3^{x}=135\times3^{x}\) (but the form is \(\frac{5}{3}(3)^{x + 4}\), when \(x = 0\), \(f(0)=\frac{5}{3}\times3^{4}=\frac{5}{3}\times81 = 135\), but the form is not showing the y - intercept as a coefficient directly in the way we need. Wait, no, let's check the requirement: the form that shows the y - coordinate of the y - intercept as a constant or coefficient. The y - intercept is at \(x = 0\), so we can also analyze each option by plugging \(x = 0\) or rewriting the function.
• Option B: \(f(x)=15(3)^{x + 2}=15\times3^{x}\times3^{2}=15\times9\times3^{x}=135\times3^{x}\)? Wait, \(15\times9 = 135\)? No, \(15\times9=135\)? Wait, \(3^{2}=9\), \(15\times9 = 135\)? Wait, no, \(15\times9 = 135\)? Wait, \(15\times9=135\), but let's rewrite \(15(3)^{x + 2}\) using exponent rule: \(15\times3^{x}\times3^{2}=15\times9\times3^{x}=135\times3^{x}\)? Wait, no, \(15\times9 = 135\), but let's check the original function. Wait, the original function is \(5(3)^{x + 3}\). Let's rewrite option B: \(15(3)^{x+2}=15\times3^{x}\times3^{2}=15\times9\times3^{x}=135\times3^{x}\)? Wait, no, \(15\times9 = 135\), but let's check the exponent. Wait, \(x + 2\) vs \(x+3\) in the original. Wait, our simplification of the original function gives \(f(x)=135\times3^{x}\), which is option D. Wait, let's do it step by step.

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

Using \(a^{m + n}=a^{m}\times a^{n}\), \(3^{x+3}=3^{x}\times3^{3}\), so \(f(x)=5\times3^{3}\times3^{x}\)

\(3^{3}=27\), \(5\times27 = 135\), so \(f(x)=135\times3^{x}\), which is option D.

Let's verify the y - intercept: For \(f(x)=135\times3^{x}\), when \(x = 0\), \(f(0)=135\times3^{0}=135\times1 = 135\). Now check the original function: \(f(0)=5\times3^{0 + 3}=5\times27 = 135\). Now check option D: \(f(x)=135\times3^{x}\), when \(x = 0\), \(f(0)=135\), and the coefficient is 135, which is the y - coordinate of the y - intercept.

Let's check other options:

Option A: \(f(x)=\frac{5}{3}(3)^{x + 4}\), 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: \(f(x)=15(3)^{x + 2}\), when \(x = 0\), \(f(0)=15\times3^{2}=15\times9 = 135\), but the form is \(15(3)^{x + 2}\), the coefficient is 15, not 135.

Option C: \(f(x)=45(3)^{x + 1}\), when \(x = 0\), \(f(0)=45\times3^{1}=45\times3 = 135\), but the coefficient is 45, not 135.

Option D: \(f(x)=135(3)^{x}\), when \(x = 0\), \(f(0)=135\times3^{0}=135\), and the coefficient is 135, which is the y - coordinate of the y - intercept. So this form shows the y - coordinate of the y - intercept as a coefficient.

Answer:

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