The task is to graph the exponential function:

$f(x) = \left(\frac{4}{3}\right)^x$

To do this, you'll need to calculate the function values $f(x)$ for specific $x$-values and plot the points. Let's break it down step-by-step:

Step 1: Choose $x$-values

Common choices for $x$ include: $-2, -1, 0, 1, 2$.

Step 2: Calculate $f(x)$

We calculate $f(x)$ for each chosen $x$:

• For $x = -2$: $f(-2) = \left(\frac{4}{3}\right)^{-2} = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \approx 0.56$
• For $x = -1$: $f(-1) = \left(\frac{4}{3}\right)^{-1} = \frac{3}{4} = 0.75$
• For $x = 0$: $f(0) = \left(\frac{4}{3}\right)^0 = 1$
• For $x = 1$: $f(1) = \left(\frac{4}{3}\right)^1 = \frac{4}{3} \approx 1.33$
• For $x = 2$: $f(2) = \left(\frac{4}{3}\right)^2 = \frac{16}{9} \approx 1.78$

Step 3: Plot Points

The five points to plot are approximately:

• $(-2, 0.56)$
• $(-1, 0.75)$
• $(0, 1)$
• $(1, 1.33)$
• $(2, 1.78)$

Step 4: Sketch the Graph

After plotting these points, draw a smooth curve that increases as $x$ increases, reflecting the growth of the exponential function $\left(\frac{4}{3}\right)^x$.

Would you like additional details on how to plot this graph or verify the calculations?

Related Questions:

1. What happens to $f(x)$ as $x$ approaches negative infinity?
2. How does $\left(\frac{4}{3}\right)^x$ differ from $2^x$ in terms of growth rate?
3. How would the graph look if the base were less than 1, e.g., $f(x) = \left(\frac{3}{4}\right)^x$?
4. Can you derive the general formula for the slope of this function at any point?
5. How does the horizontal asymptote of $f(x) = \left(\frac{4}{3}\right)^x$ affect the graph?

Tip:

In exponential functions of the form $f(x) = b^x$, the base $b > 1$ indicates growth, while $0 < b < 1$ indicates decay.