To understand the graph of the function [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex], let's analyze its key properties step-by-step:
1. Type of Function:
The given function is an exponential function of the form [tex]\( f(x) = ab^x \)[/tex], where:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = \frac{2}{3} \)[/tex].
2. Base of the Exponential Function:
Since [tex]\( \frac{2}{3} \)[/tex] is less than 1 and greater than 0, [tex]\( b = \frac{2}{3} \)[/tex] indicates that the function will exhibit exponential decay.
3. Y-Intercept:
The y-intercept of the function can be found by evaluating [tex]\( f(x) \)[/tex] at [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = 3 \left(\frac{2}{3}\right)^0 = 3 \cdot 1 = 3
\][/tex]
Thus, the point (0, 3) is on the graph.
4. Behavior as [tex]\( x \to \infty \)[/tex]:
As [tex]\( x \to \infty \)[/tex], [tex]\( \left(\frac{2}{3}\right)^x \to 0 \)[/tex] because [tex]\(\frac{2}{3}\)[/tex] is less than 1. Therefore, [tex]\( f(x) \to 0 \)[/tex].
5. Behavior as [tex]\( x \to -\infty \)[/tex]:
As [tex]\( x \to -\infty \)[/tex], [tex]\( \left(\frac{2}{3}\right)^x \to \infty \)[/tex] since the base [tex]\( \frac{2}{3} \)[/tex] raised to a large negative number becomes very large. This makes the function grow rapidly for negative values of [tex]\(x\)[/tex].
6. Graph Characteristics:
- The function decreases as [tex]\( x \)[/tex] increases, due to the base being less than 1.
- The graph approaches 0 but never actually reaches it, indicating a horizontal asymptote at [tex]\( y = 0 \)[/tex].
- The function is always positive, given that the exponential decay doesn't make it negative.
7. Key Points:
Evaluate the function at a few more key points to understand the shape:
- At [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = 3 \left( \frac{2}{3} \right)^1 = 3 \cdot \frac{2}{3} = 2
\][/tex]
So, the point (1, 2) is on the graph.
- At [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = 3 \left( \frac{2}{3} \right)^{-1} = 3 \cdot \frac{3}{2} = 4.5
\][/tex]
So, the point (-1, 4.5) is on the graph.
Given these detailed steps and properties, the graph of [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex] is an exponentially decaying curve that crosses the y-axis at (0, 3) and approaches 0 asymptotically as [tex]\( x \to \infty \)[/tex], while it increases rapidly as [tex]\( x \)[/tex] approaches negative infinity.
