Answer :
To graph the function [tex]\( f(x) = 2^{x+1} \)[/tex], follow these steps:
### Step 1: Understand the Function
The function [tex]\( f(x) = 2^{x+1} \)[/tex] is an exponential function. An exponential function of the form [tex]\( a^{x+b} \)[/tex] has these basic properties:
- The base (in this case, [tex]\( 2 \)[/tex]) is greater than 1, so the function will grow exponentially.
- The expression [tex]\( x + 1 \)[/tex] indicates a horizontal shift to the left by 1 unit from the parent function [tex]\( y = 2^x \)[/tex].
### Step 2: Determine Key Points
For plotting an exponential function, identifying some key points helps in accurately drawing the curve. Here are a few points we can use:
1. At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2^{-1+1} = 2^0 = 1 \][/tex]
Point: [tex]\( (-1, 1) \)[/tex]
2. At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2^{0+1} = 2^1 = 2 \][/tex]
Point: [tex]\( (0, 2) \)[/tex]
3. At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2^{1+1} = 2^2 = 4 \][/tex]
Point: [tex]\( (1, 4) \)[/tex]
4. At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2^{2+1} = 2^3 = 8 \][/tex]
Point: [tex]\( (2, 8) \)[/tex]
5. At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 2^{-2+1} = 2^{-1} = \frac{1}{2} \][/tex]
Point: [tex]\( (-2, 0.5) \)[/tex]
### Step 3: Plotting the Key Points
1. [tex]\( (-2, 0.5) \)[/tex]
2. [tex]\( (-1, 1) \)[/tex]
3. [tex]\( (0, 2) \)[/tex]
4. [tex]\( (1, 4) \)[/tex]
5. [tex]\( (2, 8) \)[/tex]
### Step 4: Draw the Asymptote
The horizontal asymptote of this exponential function [tex]\( f(x) = 2^{x+1} \)[/tex] is the x-axis (or [tex]\( y = 0 \)[/tex]) because, as [tex]\( x \)[/tex] approaches negative infinity, the value of [tex]\( 2^{x+1} \)[/tex] approaches 0.
### Step 5: Sketch the Graph
Now that we have the key points and the asymptote, we can sketch the graph:
1. Plot the points mentioned above on a coordinate plane.
2. Draw a smooth, curved line through these points. Make sure to approach the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] moves towards negative infinity, and let the function grow rapidly for positive values of [tex]\( x \)[/tex].
### Visualization
Here is a rough sketch of the function [tex]\( f(x) = 2^{x+1} \)[/tex]:
```
|
8 | (2, 8)
|
4 | (1, 4)
|
2 | (0, 2)
|
1 -----------------------|----------- x
| (-1, 1)
0.5 | * (-2, 0.5)
|
0---------------------------------------------------
-2 -1 0 1 2
```
Note how the curve gets closer to the x-axis (y=0) as [tex]\( x \)[/tex] becomes more negative, and grows rapidly as [tex]\( x \)[/tex] increases.
By following these steps and plotting these key points, you effectively graph the exponential function [tex]\( f(x) = 2^{x+1} \)[/tex].
### Step 1: Understand the Function
The function [tex]\( f(x) = 2^{x+1} \)[/tex] is an exponential function. An exponential function of the form [tex]\( a^{x+b} \)[/tex] has these basic properties:
- The base (in this case, [tex]\( 2 \)[/tex]) is greater than 1, so the function will grow exponentially.
- The expression [tex]\( x + 1 \)[/tex] indicates a horizontal shift to the left by 1 unit from the parent function [tex]\( y = 2^x \)[/tex].
### Step 2: Determine Key Points
For plotting an exponential function, identifying some key points helps in accurately drawing the curve. Here are a few points we can use:
1. At [tex]\( x = -1 \)[/tex]:
[tex]\[ f(-1) = 2^{-1+1} = 2^0 = 1 \][/tex]
Point: [tex]\( (-1, 1) \)[/tex]
2. At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 2^{0+1} = 2^1 = 2 \][/tex]
Point: [tex]\( (0, 2) \)[/tex]
3. At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 2^{1+1} = 2^2 = 4 \][/tex]
Point: [tex]\( (1, 4) \)[/tex]
4. At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2^{2+1} = 2^3 = 8 \][/tex]
Point: [tex]\( (2, 8) \)[/tex]
5. At [tex]\( x = -2 \)[/tex]:
[tex]\[ f(-2) = 2^{-2+1} = 2^{-1} = \frac{1}{2} \][/tex]
Point: [tex]\( (-2, 0.5) \)[/tex]
### Step 3: Plotting the Key Points
1. [tex]\( (-2, 0.5) \)[/tex]
2. [tex]\( (-1, 1) \)[/tex]
3. [tex]\( (0, 2) \)[/tex]
4. [tex]\( (1, 4) \)[/tex]
5. [tex]\( (2, 8) \)[/tex]
### Step 4: Draw the Asymptote
The horizontal asymptote of this exponential function [tex]\( f(x) = 2^{x+1} \)[/tex] is the x-axis (or [tex]\( y = 0 \)[/tex]) because, as [tex]\( x \)[/tex] approaches negative infinity, the value of [tex]\( 2^{x+1} \)[/tex] approaches 0.
### Step 5: Sketch the Graph
Now that we have the key points and the asymptote, we can sketch the graph:
1. Plot the points mentioned above on a coordinate plane.
2. Draw a smooth, curved line through these points. Make sure to approach the horizontal asymptote [tex]\( y = 0 \)[/tex] as [tex]\( x \)[/tex] moves towards negative infinity, and let the function grow rapidly for positive values of [tex]\( x \)[/tex].
### Visualization
Here is a rough sketch of the function [tex]\( f(x) = 2^{x+1} \)[/tex]:
```
|
8 | (2, 8)
|
4 | (1, 4)
|
2 | (0, 2)
|
1 -----------------------|----------- x
| (-1, 1)
0.5 | * (-2, 0.5)
|
0---------------------------------------------------
-2 -1 0 1 2
```
Note how the curve gets closer to the x-axis (y=0) as [tex]\( x \)[/tex] becomes more negative, and grows rapidly as [tex]\( x \)[/tex] increases.
By following these steps and plotting these key points, you effectively graph the exponential function [tex]\( f(x) = 2^{x+1} \)[/tex].
