Graph each exponential function. 5. $$ y=3^{x} $$

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UpStudy AI · Accepted Answer

Sure! Let's graph the exponential function:

$$ y = 3^{x} $$

### Understanding the Function

- **Base:** The base of the exponential function is **3**, which is greater than 1. This means the function will **increase rapidly** as $$ x $$ increases.
- **Domain:** All real numbers ($$ -\infty < x < \infty $$)
- **Range:** All positive real numbers ($$ 0 < y < \infty $$)
- **Y-Intercept:** $$ (0, 1) $$ because $$ 3^0 = 1 $$
- **Horizontal Asymptote:** $$ y = 0 $$ (the graph approaches but never touches the x-axis)
- **Behavior:**
  - As $$ x $$ increases, $$ y $$ increases rapidly.
  - As $$ x $$ decreases (approaches negative infinity), $$ y $$ approaches 0.

### Table of Values

| $$ x $$ | $$ y = 3^{x} $$ |
|---------|-----------------|
| -2      | $$ \frac{1}{9} \approx 0.111 $$ |
| -1      | $$ \frac{1}{3} \approx 0.333 $$ |
| 0       | 1               |
| 1       | 3               |
| 2       | 9               |
| 3       | 27              |

### Graph of $$ y = 3^{x} $$

Below is an approximate ASCII representation of the graph:

```
y
|
|                                        *
|                                     *
|                                 *
|                             *
|                         *
|                     *
|                 *
|             *
|          *
|      *
|   *
|*_______________________________ x
```

### Description of the Graph

1. **Point (0, 1):** The graph passes through the point where $$ x = 0 $$ and $$ y = 1 $$.
2. **Rapid Growth:** As $$ x $$ increases, the value of $$ y $$ increases exponentially. For example, when $$ x = 3 $$, $$ y = 27 $$.
3. **Approaching the X-Axis:** As $$ x $$ decreases, $$ y $$ approaches 0 but never actually reaches it, illustrating the horizontal asymptote at $$ y = 0 $$.
4. **Increasing Function:** Since the base $$ 3 > 1 $$, the function is always increasing; there are no maximum or minimum points other than the y-intercept.

### Plotting the Graph with Software

For a more accurate and visually detailed graph, you can use graphing tools such as:

- **Desmos:** [https://www.desmos.com/calculator](https://www.desmos.com/calculator)
- **GeoGebra:** [https://www.geogebra.org/graphing](https://www.geogebra.org/graphing)
- **Graphing Calculators:** Physical or online calculators that support graphing functions.

Simply enter the function $$ y = 3^{x} $$ into any of these tools to see a precise graph.

### Summary

The exponential function $$ y = 3^{x} $$ is an increasing function that grows rapidly as $$ x $$ increases. It has a y-intercept at (0, 1) and approaches the x-axis as $$ x $$ becomes more negative. This function is useful in various real-world applications, such as compound interest, population growth, and more.

If you have any specific questions about this function or need further assistance, feel free to ask!
To graph the exponential function $$ y = 3^{x} $$: 1. **Plot the Y-Intercept:** Start at (0, 1). 2. **Plot Additional Points:** For example, at $$ x = 1 $$, $$ y = 3 $$; at $$ x = 2 $$, $$ y = 9 $$. 3. **Draw the Curve:** Connect the points with a smooth curve that increases rapidly as $$ x $$ increases. 4. **Add Asymptote:** Draw a horizontal line at $$ y = 0 $$ to represent the horizontal asymptote. The graph will show an increasing curve starting at (0,1) and rising steeply as $$ x $$ increases, approaching the x-axis but never touching it.

Graph each exponential function.
5.

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