Answer :
To determine the graph of the function [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex], let's analyze the behavior of the function and calculate a few key points:
1. Intercepts:
- y-intercept: The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \left( \frac{2}{3} \right)^0 = 3 \cdot 1 = 3 \][/tex]
So, the y-intercept is at [tex]\( (0, 3) \)[/tex].
- x-intercept: To find the x-intercept, we set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ 3\left(\frac{2}{3}\right)^x = 0 \][/tex]
However, since [tex]\( 3\left(\frac{2}{3}\right)^x \)[/tex] never actually equals zero for any real value of [tex]\( x \)[/tex], there is no x-intercept.
2. Behavior as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \)[/tex] increases ([tex]\( x \to \infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^x \to 0 \)[/tex], hence [tex]\( f(x) \to 0 \)[/tex].
- As [tex]\( x \)[/tex] decreases ([tex]\( x \to -\infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] grows exponentially because the base [tex]\( \frac{2}{3} < 1 \)[/tex]. Consequently, [tex]\( f(x) \to \infty \)[/tex].
3. Key values to plot:
Let's find some values around [tex]\( x = 0 \)[/tex] to get a sense for how the function behaves for other points:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3\left(\frac{2}{3}\right)^{-1} = 3 \cdot \frac{3}{2} = 4.5 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3\left(\frac{2}{3}\right) = 3 \cdot \frac{2}{3} = 2 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3\left(\frac{2}{3}\right)^2 = 3 \cdot \left(\frac{4}{9}\right) = \frac{12}{9} = \frac{4}{3} \approx 1.33 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3\left(\frac{2}{3}\right)^{-2} = 3 \cdot \left(\frac{3}{2}\right)^2 = 3 \cdot \frac{9}{4} = 6.75 \][/tex]
4. A sample of points:
Here are some approximate points describing the function behavior close to the input:
[tex]\[ \begin{aligned} (-10, 172.99), & \quad (-9.95, 169.51), \quad (-9.90, 166.10), \\ (-9.85, 162.76), & \quad (-9.80, 159.49), \quad (-9.75, 156.28), \\ (-9.70, 153.13), & \quad (-9.65, 150.05), \quad (-9.60, 147.03), \\ (-9.55, 144.08). \end{aligned} \][/tex]
Given the points we have calculated and the behavior described, we can draw the graph of [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex]:
- The function will decay towards 0 as [tex]\( x \to \infty \)[/tex].
- The function will escalate to large values as [tex]\( x \to -\infty \)[/tex].
- The y-intercept is at [tex]\( (0, 3) \)[/tex].
The general shape of the graph will show exponential decay towards the positive x-axis with a rapid increase when moving towards the negative x-axis.
1. Intercepts:
- y-intercept: The y-intercept occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3 \left( \frac{2}{3} \right)^0 = 3 \cdot 1 = 3 \][/tex]
So, the y-intercept is at [tex]\( (0, 3) \)[/tex].
- x-intercept: To find the x-intercept, we set [tex]\( f(x) = 0 \)[/tex]:
[tex]\[ 3\left(\frac{2}{3}\right)^x = 0 \][/tex]
However, since [tex]\( 3\left(\frac{2}{3}\right)^x \)[/tex] never actually equals zero for any real value of [tex]\( x \)[/tex], there is no x-intercept.
2. Behavior as [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex]:
- As [tex]\( x \)[/tex] increases ([tex]\( x \to \infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^x \to 0 \)[/tex], hence [tex]\( f(x) \to 0 \)[/tex].
- As [tex]\( x \)[/tex] decreases ([tex]\( x \to -\infty \)[/tex]), [tex]\( \left(\frac{2}{3}\right)^x \)[/tex] grows exponentially because the base [tex]\( \frac{2}{3} < 1 \)[/tex]. Consequently, [tex]\( f(x) \to \infty \)[/tex].
3. Key values to plot:
Let's find some values around [tex]\( x = 0 \)[/tex] to get a sense for how the function behaves for other points:
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 3\left(\frac{2}{3}\right)^{-1} = 3 \cdot \frac{3}{2} = 4.5 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 3\left(\frac{2}{3}\right) = 3 \cdot \frac{2}{3} = 2 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 3\left(\frac{2}{3}\right)^2 = 3 \cdot \left(\frac{4}{9}\right) = \frac{12}{9} = \frac{4}{3} \approx 1.33 \][/tex]
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 3\left(\frac{2}{3}\right)^{-2} = 3 \cdot \left(\frac{3}{2}\right)^2 = 3 \cdot \frac{9}{4} = 6.75 \][/tex]
4. A sample of points:
Here are some approximate points describing the function behavior close to the input:
[tex]\[ \begin{aligned} (-10, 172.99), & \quad (-9.95, 169.51), \quad (-9.90, 166.10), \\ (-9.85, 162.76), & \quad (-9.80, 159.49), \quad (-9.75, 156.28), \\ (-9.70, 153.13), & \quad (-9.65, 150.05), \quad (-9.60, 147.03), \\ (-9.55, 144.08). \end{aligned} \][/tex]
Given the points we have calculated and the behavior described, we can draw the graph of [tex]\( f(x) = 3\left(\frac{2}{3}\right)^x \)[/tex]:
- The function will decay towards 0 as [tex]\( x \to \infty \)[/tex].
- The function will escalate to large values as [tex]\( x \to -\infty \)[/tex].
- The y-intercept is at [tex]\( (0, 3) \)[/tex].
The general shape of the graph will show exponential decay towards the positive x-axis with a rapid increase when moving towards the negative x-axis.
