To identify the graph of the function [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex], let's analyze its key characteristics step-by-step.
1. Function Type and Base:
- This function is an exponential decay function because the base [tex]\(\frac{2}{3}\)[/tex] is between 0 and 1. Therefore, as [tex]\(x\)[/tex] increases, the function [tex]\(f(x)\)[/tex] will decrease.
2. Y-intercept:
- To find the y-intercept, we evaluate [tex]\(f(x)\)[/tex] when [tex]\(x = 0\)[/tex].
[tex]\[
f(0) = 3\left(\frac{2}{3}\right)^0 = 3 \cdot 1 = 3
\][/tex]
- The point [tex]\((0,3)\)[/tex] is on the graph.
3. Behavior as [tex]\(x\)[/tex] approaches positive and negative infinity:
- As [tex]\(x \to \infty\)[/tex], [tex]\(\left(\frac{2}{3}\right)^x \to 0\)[/tex]. Therefore, [tex]\(f(x) = 3\left(\frac{2}{3}\right)^x \to 0\)[/tex].
- As [tex]\(x \to -\infty\)[/tex], [tex]\(\left(\frac{2}{3}\right)^x \to \infty\)[/tex]. Therefore, [tex]\(f(x) = 3\left(\frac{2}{3}\right)^x \to 3 \cdot \infty\)[/tex].
4. Key points:
- We already found the y-intercept at [tex]\((0, 3)\)[/tex].
- Calculate [tex]\(f(x)\)[/tex] for a few integer values of [tex]\(x\)[/tex].
[tex]\[
f(1) = 3\left(\frac{2}{3}\right)^1 = 3 \cdot \frac{2}{3} = 2
\][/tex]
- The point [tex]\((1, 2)\)[/tex] is on the graph.
[tex]\[
f(-1) = 3\left(\frac{2}{3}\right)^{-1} = 3 \cdot \frac{3}{2} = 4.5
\][/tex]
- The point [tex]\((-1, 4.5)\)[/tex] is on the graph.
5. General shape:
- The graph starts at 3 when [tex]\(x = 0\)[/tex] and decreases towards 0 as [tex]\(x\)[/tex] increases.
- As [tex]\(x\)[/tex] decreases to the left, the function value increases rapidly.
By plotting these points ([tex]\((0, 3)\)[/tex], [tex]\((1, 2)\)[/tex], [tex]\((-1, 4.5)\)[/tex]) on a coordinate plane and sketching the curve through them with the described properties (decreasing from left to right, approaching zero as [tex]\(x \to \infty\)[/tex] and increasing as [tex]\(x \to -\infty\)[/tex]), we get the graphical representation of the function [tex]\(f(x) = 3\left(\frac{2}{3}\right)^x\)[/tex].
