Question

tical stretches and shrinks of exponential functions
which exponential function has an initial value of 3?

$f(x) = \frac{1}{3}(9)^x$

$f(x) = (3)^x$

$x$	$f(x)$
-1	-3.75
0	-3
1	0
2	12

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = a(b)^x \), where \( a \) is the initial value (when \( x = 0 \), \( f(0)=a \)).

Step2: Analyze each option

• For \( f(x)=\frac{1}{3}(9)^x \): When \( x = 0 \), \( f(0)=\frac{1}{3}(9)^0=\frac{1}{3}(1)=\frac{1}{3}

eq3 \).

• For \( f(x)=(3)^x \): When \( x = 0 \), \( f(0)=3^0 = 1

eq3 \) (this was marked wrong, which matches).

• Now check the graph: The graph of an exponential function with initial value 3 should pass through \( (0, 3) \). Looking at the first graph, when \( x = 0 \), the \( y \)-value (initial value) appears to be 3 (visually, the curve crosses the \( y \)-axis at \( y = 3 \)). Also, let's re - evaluate the first function's graph. The graph shown (first box) has a \( y \)-intercept (initial value) of 3. Wait, maybe we mis - checked the functions. Wait, maybe there was a mistake in the previous analysis. Wait, let's re - check the functions. Wait, the user's question is which exponential function has an initial value of 3. Let's re - examine the functions:

Wait, maybe the first graph corresponds to a function. Wait, the two functions given: \( f(x)=\frac{1}{3}(9)^x \) and \( f(x)=3^x \) were wrong. Wait, maybe the correct function is the one whose graph has an initial value of 3. Wait, the initial value is when \( x = 0 \). Let's re - calculate for the first function: no, wait, maybe the graph is the correct one. Wait, the problem is to find which exponential function has an initial value of 3. The initial value is \( f(0) \). Let's assume the graph is a function like \( f(x)=3(b)^x \), which has an initial value of 3. But among the given functions, maybe there was a mistake in the options. Wait, no, the user's image has a graph, a function \( \frac{1}{3}(9)^x \), and \( 3^x \) (marked wrong). Wait, maybe the correct answer is the graph (the first box) or maybe we made a mistake in the function analysis. Wait, no, let's re - do the function analysis.

Wait, the general form is \( f(x)=a\cdot b^x \), initial value \( a \) (at \( x = 0 \)). Let's check the graph: the graph in the first box, when \( x = 0 \), the \( y \)-coordinate is 3 (visually, the curve is at \( y = 3 \) when \( x = 0 \)). So the graph (the first box) represents an exponential function with initial value 3.

Answer:

The graph in the first box (the left - most graph)