To graph the exponential function [tex]\( f(x) = \left(\frac{5}{3}\right)^x \)[/tex], we will first identify a few key points on the function and then determine the horizontal asymptote. Here is a detailed, step-by-step solution:
### 1. Identify the Function
The given function is
[tex]\[
f(x) = \left(\frac{5}{3}\right)^x
\][/tex]
### 2. Generate Points
We'll select five values of [tex]\( x \)[/tex] and calculate the corresponding [tex]\( y \)[/tex] values using our function. Let's choose [tex]\( x = -2, -1, 0, 1, 2 \)[/tex].
- For [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) = \left(\frac{5}{3}\right)^{-2} \approx 0.36
\][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) = \left(\frac{5}{3}\right)^{-1} \approx 0.6
\][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) = \left(\frac{5}{3}\right)^0 = 1
\][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) = \left(\frac{5}{3}\right)^1 \approx 1.67
\][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) = \left(\frac{5}{3}\right)^2 \approx 2.78
\][/tex]
So, the points we have now are:
[tex]\[
(-2, 0.36), (-1, 0.6), (0, 1), (1, 1.67), (2, 2.78)
\][/tex]
### 3. Determine the Asymptote
For the exponential function [tex]\( f(x) = \left(\frac{5}{3}\right)^x \)[/tex], the horizontal asymptote is the line [tex]\( y = 0 \)[/tex]. This is because as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( \left( \frac{5}{3} \right)^x \)[/tex] approaches zero but never actually reaches it.
### 4. Plot the Points and the Asymptote
Now, let's plot the points and the asymptote on a graph:
- Plot the points [tex]\( (-2, 0.36) \)[/tex], [tex]\( (-1, 0.6) \)[/tex], [tex]\( (0, 1) \)[/tex], [tex]\( (1, 1.67) \)[/tex], and [tex]\( (2, 2.78) \)[/tex].
- Draw a horizontal line at [tex]\( y = 0 \)[/tex] to represent the asymptote.
### Graph Representation
Here is a simplified version of what the graph looks like (you would need to construct this on graph paper or using graphing software for accuracy):
```
y
|
3 |
|
2 |
|
1 |
|
|
0 |------------------------------------------------ x
-2 -1 0 1 2
```
- Where each '' represents a plotted point.
- The horizontal line along [tex]\( y = 0 \)[/tex] represents the asymptote (not drawn precisely in text format, but you should include it on graph paper or software).
### Conclusion
We have successfully plotted the points on the graph: [tex]\( (-2, 0.36), (-1, 0.6), (0, 1), (1, 1.67), (2, 2.78) \)[/tex]. The horizontal asymptote for the function [tex]\( f(x) = \left(\frac{5}{3}\right)^x \)[/tex] is [tex]\( y = 0 \)[/tex]. Now you can use the graph-a-function button in your graphing tool to visualize the function smoothly.
