Answer :
Certainly! Let's go through the process of graphing the function [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex] step by step:
### Understanding the Function
First, let's understand the given function:
[tex]\[ g(x) = \frac{1}{2} \cdot 2^x \][/tex]
This can be rewritten to make it clearer:
[tex]\[ g(x) = \frac{1}{2} \cdot 2^x = 2^{x-1} \][/tex]
#### Key Points
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = \frac{1}{2} \cdot 2^0 = \frac{1}{2} \cdot 1 = \frac{1}{2} \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{1}{2} \cdot 2^1 = \frac{1}{2} \cdot 2 = 1 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{1}{2} \cdot 2^{-1} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{1}{2} \cdot 2^2 = \frac{1}{2} \cdot 4 = 2 \][/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to 0 \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
### Creating a Table of Values
Let's create a table of values for [tex]\( g(x) \)[/tex]:
| [tex]\( x \)[/tex] | [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex] |
|--------|---------------------------------|
| -3 | 0.0625 |
| -2 | 0.125 |
| -1 | 0.25 |
| 0 | 0.5 |
| 1 | 1 |
| 2 | 2 |
| 3 | 4 |
| 4 | 8 |
### Plotting the Graph
Now, let's plot the points from our table and sketch the graph of [tex]\( g(x) \)[/tex]:
1. Begin by drawing the coordinate axes.
2. Mark each of the calculated points in the table on the graph.
3. Connect the points with a smooth curve.
### Key Characteristics of the Graph
- The graph will pass through the points we calculated.
- As [tex]\( x \to -\infty \)[/tex], the graph approaches the x-axis (i.e., [tex]\( y = 0 \)[/tex]).
- As [tex]\( x \to +\infty \)[/tex], the graph rises sharply.
- The graph is an exponential curve because it involves [tex]\( 2^x \)[/tex].
### Final Graph
The graph of [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex] is an exponentially increasing function. More formally, the graph:
- Starts very close to the x-axis for large negative [tex]\( x \)[/tex].
- Passes through the point [tex]\( (0, 0.5) \)[/tex].
- Rises rapidly for positive [tex]\( x \)[/tex], crossing points like [tex]\( (1, 1) \)[/tex] and [tex]\( (2, 2) \)[/tex].
Here is a rough sketch of how it should look:
```
8 +
|
|
4 +
|
|
2 +
|
1 +
|
0.5 +
|
0 +-------------------------------+-----> x
-3 -2 -1 0 1 2 3
```
This sketch demonstrates the general shape and behavior of the function [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex].
### Understanding the Function
First, let's understand the given function:
[tex]\[ g(x) = \frac{1}{2} \cdot 2^x \][/tex]
This can be rewritten to make it clearer:
[tex]\[ g(x) = \frac{1}{2} \cdot 2^x = 2^{x-1} \][/tex]
#### Key Points
- When [tex]\( x = 0 \)[/tex]:
[tex]\[ g(0) = \frac{1}{2} \cdot 2^0 = \frac{1}{2} \cdot 1 = \frac{1}{2} \][/tex]
- When [tex]\( x = 1 \)[/tex]:
[tex]\[ g(1) = \frac{1}{2} \cdot 2^1 = \frac{1}{2} \cdot 2 = 1 \][/tex]
- When [tex]\( x = -1 \)[/tex]:
[tex]\[ g(-1) = \frac{1}{2} \cdot 2^{-1} = \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4} \][/tex]
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ g(2) = \frac{1}{2} \cdot 2^2 = \frac{1}{2} \cdot 4 = 2 \][/tex]
- As [tex]\( x \to -\infty \)[/tex], [tex]\( g(x) \to 0 \)[/tex].
- As [tex]\( x \to +\infty \)[/tex], [tex]\( g(x) \to \infty \)[/tex].
### Creating a Table of Values
Let's create a table of values for [tex]\( g(x) \)[/tex]:
| [tex]\( x \)[/tex] | [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex] |
|--------|---------------------------------|
| -3 | 0.0625 |
| -2 | 0.125 |
| -1 | 0.25 |
| 0 | 0.5 |
| 1 | 1 |
| 2 | 2 |
| 3 | 4 |
| 4 | 8 |
### Plotting the Graph
Now, let's plot the points from our table and sketch the graph of [tex]\( g(x) \)[/tex]:
1. Begin by drawing the coordinate axes.
2. Mark each of the calculated points in the table on the graph.
3. Connect the points with a smooth curve.
### Key Characteristics of the Graph
- The graph will pass through the points we calculated.
- As [tex]\( x \to -\infty \)[/tex], the graph approaches the x-axis (i.e., [tex]\( y = 0 \)[/tex]).
- As [tex]\( x \to +\infty \)[/tex], the graph rises sharply.
- The graph is an exponential curve because it involves [tex]\( 2^x \)[/tex].
### Final Graph
The graph of [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex] is an exponentially increasing function. More formally, the graph:
- Starts very close to the x-axis for large negative [tex]\( x \)[/tex].
- Passes through the point [tex]\( (0, 0.5) \)[/tex].
- Rises rapidly for positive [tex]\( x \)[/tex], crossing points like [tex]\( (1, 1) \)[/tex] and [tex]\( (2, 2) \)[/tex].
Here is a rough sketch of how it should look:
```
8 +
|
|
4 +
|
|
2 +
|
1 +
|
0.5 +
|
0 +-------------------------------+-----> x
-3 -2 -1 0 1 2 3
```
This sketch demonstrates the general shape and behavior of the function [tex]\( g(x) = \frac{1}{2} \cdot 2^x \)[/tex].
