At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Certainly! Let's go through the process of graphing the exponential function [tex]\( w(x) = 2 \cdot (0.4)^x - 5 \)[/tex] step by step.
### Step 1: Understand the Function
The given function is an exponential decay function because the base of the exponent [tex]\(0.4\)[/tex] is between 0 and 1. Here is the breakdown of the function:
- [tex]\( 2 \)[/tex] is the coefficient that vertically stretches the graph.
- [tex]\( 0.4 \)[/tex] is the base of the exponent, causing decay.
- [tex]\( x \)[/tex] is the variable exponent.
- [tex]\( -5 \)[/tex] is a vertical shift downward by 5 units.
### Step 2: Plot Key Points
To graph the function, let's compute some key points.
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[ w(0) = 2 \cdot (0.4)^0 - 5 = 2 \cdot 1 - 5 = 2 - 5 = -3 \][/tex]
Point: [tex]\((0, -3)\)[/tex]
2. When [tex]\( x = 1 \)[/tex]:
[tex]\[ w(1) = 2 \cdot (0.4)^1 - 5 = 2 \cdot 0.4 - 5 = 0.8 - 5 = -4.2 \][/tex]
Point: [tex]\((1, -4.2)\)[/tex]
3. When [tex]\( x = -1 \)[/tex]:
[tex]\[ w(-1) = 2 \cdot (0.4)^{-1} - 5 = 2 \cdot \frac{1}{0.4} - 5 = 2 \cdot 2.5 - 5 = 5 - 5 = 0 \][/tex]
Point: [tex]\((-1, 0)\)[/tex]
4. When [tex]\( x = 2 \)[/tex]:
[tex]\[ w(2) = 2 \cdot (0.4)^2 - 5 = 2 \cdot 0.16 - 5 = 0.32 - 5 = -4.68 \][/tex]
Point: [tex]\((2, -4.68)\)[/tex]
5. When [tex]\( x = -2 \)[/tex]:
[tex]\[ w(-2) = 2 \cdot (0.4)^{-2} - 5 = 2 \cdot \left(\frac{1}{0.4}\right)^2 - 5 = 2 \cdot 6.25 - 5 = 12.5 - 5 = 7.5 \][/tex]
Point: [tex]\((-2, 7.5)\)[/tex]
### Step 3: Additional Points
You can calculate more points if needed to provide better accuracy. Here are additional common values:
#### When [tex]\( x = -3 \)[/tex]:
[tex]\[ w(-3) = 2 \cdot (0.4)^{-3} - 5 = 2 \cdot \left(\frac{1}{0.4}\right)^3 - 5 = 2 \cdot 15.625 - 5 = 31.25 - 5 = 26.25 \][/tex]
Point: [tex]\((-3, 26.25)\)[/tex]
#### When [tex]\( x = 3 \)[/tex]:
[tex]\[ w(3) = 2 \cdot (0.4)^3 - 5 = 2 \cdot 0.064 - 5 = 0.128 - 5 = -4.872 \][/tex]
Point: [tex]\((3, -4.872)\)[/tex]
### Step 4: Plot the Graph
1. Plot the calculated points on a coordinate plane:
- [tex]\((0, -3)\)[/tex]
- [tex]\((1, -4.2)\)[/tex]
- [tex]\((-1, 0)\)[/tex]
- [tex]\((2, -4.68)\)[/tex]
- [tex]\((-2, 7.5)\)[/tex]
- [tex]\((-3, 26.25)\)[/tex]
- [tex]\((3, -4.872)\)[/tex]
2. Draw a smooth curve through these points, keeping in mind the overall shape of an exponential decay function: the graph should approach the horizontal line [tex]\( y = -5 \)[/tex] as [tex]\( x \)[/tex] increases, but never touch it. For negative values of [tex]\( x \)[/tex], the function increases rapidly.
### Step 5: Add Labels and Title
- Label the x-axis and y-axis appropriately.
- Title the graph: "Graph of [tex]\( w(x) = 2 \cdot (0.4)^x - 5 \)[/tex]"
### Summary
By completing these steps, you will have a clear graph representing the function [tex]\( w(x) = 2 \cdot (0.4)^x - 5 \)[/tex], which displays its exponential decay nature and vertical displacement.
### Example Graph (Not to scale):
```
y
^
|
10+
|
5+
|
0+-----*------------->
| (0,-3) x
- 5+------------------
|
```
### Conclusion
This graph showcases the behavior of the exponential decay function, with a vertical shift downward by 5 units, and how it approaches the horizontal asymptote at [tex]\( y = -5 \)[/tex].
### Step 1: Understand the Function
The given function is an exponential decay function because the base of the exponent [tex]\(0.4\)[/tex] is between 0 and 1. Here is the breakdown of the function:
- [tex]\( 2 \)[/tex] is the coefficient that vertically stretches the graph.
- [tex]\( 0.4 \)[/tex] is the base of the exponent, causing decay.
- [tex]\( x \)[/tex] is the variable exponent.
- [tex]\( -5 \)[/tex] is a vertical shift downward by 5 units.
### Step 2: Plot Key Points
To graph the function, let's compute some key points.
1. When [tex]\( x = 0 \)[/tex]:
[tex]\[ w(0) = 2 \cdot (0.4)^0 - 5 = 2 \cdot 1 - 5 = 2 - 5 = -3 \][/tex]
Point: [tex]\((0, -3)\)[/tex]
2. When [tex]\( x = 1 \)[/tex]:
[tex]\[ w(1) = 2 \cdot (0.4)^1 - 5 = 2 \cdot 0.4 - 5 = 0.8 - 5 = -4.2 \][/tex]
Point: [tex]\((1, -4.2)\)[/tex]
3. When [tex]\( x = -1 \)[/tex]:
[tex]\[ w(-1) = 2 \cdot (0.4)^{-1} - 5 = 2 \cdot \frac{1}{0.4} - 5 = 2 \cdot 2.5 - 5 = 5 - 5 = 0 \][/tex]
Point: [tex]\((-1, 0)\)[/tex]
4. When [tex]\( x = 2 \)[/tex]:
[tex]\[ w(2) = 2 \cdot (0.4)^2 - 5 = 2 \cdot 0.16 - 5 = 0.32 - 5 = -4.68 \][/tex]
Point: [tex]\((2, -4.68)\)[/tex]
5. When [tex]\( x = -2 \)[/tex]:
[tex]\[ w(-2) = 2 \cdot (0.4)^{-2} - 5 = 2 \cdot \left(\frac{1}{0.4}\right)^2 - 5 = 2 \cdot 6.25 - 5 = 12.5 - 5 = 7.5 \][/tex]
Point: [tex]\((-2, 7.5)\)[/tex]
### Step 3: Additional Points
You can calculate more points if needed to provide better accuracy. Here are additional common values:
#### When [tex]\( x = -3 \)[/tex]:
[tex]\[ w(-3) = 2 \cdot (0.4)^{-3} - 5 = 2 \cdot \left(\frac{1}{0.4}\right)^3 - 5 = 2 \cdot 15.625 - 5 = 31.25 - 5 = 26.25 \][/tex]
Point: [tex]\((-3, 26.25)\)[/tex]
#### When [tex]\( x = 3 \)[/tex]:
[tex]\[ w(3) = 2 \cdot (0.4)^3 - 5 = 2 \cdot 0.064 - 5 = 0.128 - 5 = -4.872 \][/tex]
Point: [tex]\((3, -4.872)\)[/tex]
### Step 4: Plot the Graph
1. Plot the calculated points on a coordinate plane:
- [tex]\((0, -3)\)[/tex]
- [tex]\((1, -4.2)\)[/tex]
- [tex]\((-1, 0)\)[/tex]
- [tex]\((2, -4.68)\)[/tex]
- [tex]\((-2, 7.5)\)[/tex]
- [tex]\((-3, 26.25)\)[/tex]
- [tex]\((3, -4.872)\)[/tex]
2. Draw a smooth curve through these points, keeping in mind the overall shape of an exponential decay function: the graph should approach the horizontal line [tex]\( y = -5 \)[/tex] as [tex]\( x \)[/tex] increases, but never touch it. For negative values of [tex]\( x \)[/tex], the function increases rapidly.
### Step 5: Add Labels and Title
- Label the x-axis and y-axis appropriately.
- Title the graph: "Graph of [tex]\( w(x) = 2 \cdot (0.4)^x - 5 \)[/tex]"
### Summary
By completing these steps, you will have a clear graph representing the function [tex]\( w(x) = 2 \cdot (0.4)^x - 5 \)[/tex], which displays its exponential decay nature and vertical displacement.
### Example Graph (Not to scale):
```
y
^
|
10+
|
5+
|
0+-----*------------->
| (0,-3) x
- 5+------------------
|
```
### Conclusion
This graph showcases the behavior of the exponential decay function, with a vertical shift downward by 5 units, and how it approaches the horizontal asymptote at [tex]\( y = -5 \)[/tex].
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
