Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
Sure! Let's draw the graph of the function [tex]\( f(x) = 5^{x-2} + 2 \)[/tex] step by step:
### Step 1: Understanding the Function
The given function [tex]\( f(x) = 5^{x-2} + 2 \)[/tex] is an exponential function. The general form of an exponential function is [tex]\( f(x) = a^{b(x-c)} + d \)[/tex], where
- [tex]\( a \)[/tex] is the base of the exponential function (5 in this case),
- [tex]\( b \)[/tex] determines the growth rate,
- [tex]\( c \)[/tex] is the horizontal shift,
- [tex]\( d \)[/tex] is the vertical shift.
For [tex]\( f(x) = 5^{x-2} + 2 \)[/tex]:
- [tex]\( 5 \)[/tex] is the base,
- The exponent [tex]\( x-2 \)[/tex] indicates a horizontal shift of 2 units to the right,
- The constant [tex]\( +2 \)[/tex] represents a vertical shift of 2 units upward.
### Step 2: Identify Key Points
We can find some key points on the graph by plugging in different values of [tex]\( x \)[/tex]:
1. At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 5^{0-2} + 2 = 5^{-2} + 2 = \frac{1}{25} + 2 \approx 2.04 \][/tex]
2. At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 5^{2-2} + 2 = 5^0 + 2 = 1 + 2 = 3 \][/tex]
3. At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 5^{4-2} + 2 = 5^2 + 2 = 25 + 2 = 27 \][/tex]
4. At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 5^{1-2} + 2 = 5^{-1} + 2 = \frac{1}{5} + 2 = 2.2 \][/tex]
5. At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 5^{3-2} + 2 = 5^1 + 2 = 5 + 2 = 7 \][/tex]
### Step 3: Sketching the Graph
1. Plot the points computed in Step 2 on a coordinate plane.
- [tex]\( (0, 2.04) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (4, 27) \)[/tex]
- [tex]\( (1, 2.2) \)[/tex]
- [tex]\( (3, 7) \)[/tex]
2. Draw the curve through the points. Remember the nature of the exponential function:
- The function approaches [tex]\( y = 2 \)[/tex] (the vertical shift) as [tex]\( x \)[/tex] becomes more negative, but never actually reaches it. This is the horizontal asymptote.
- The function will rise steeply as [tex]\( x \)[/tex] increases since the base of the exponent (5) is greater than 1.
### Step 4: Analyzing the Graph
- Horizontal Asymptote: The function [tex]\( f(x) = 5^{x-2} + 2 \)[/tex] has a horizontal asymptote at [tex]\( y = 2 \)[/tex].
- Behavior:
- For [tex]\( x < 2 \)[/tex], the function value is close to 2.
- For [tex]\( x > 2 \)[/tex], the function value increases rapidly.
### Final Sketch:
Here's a rough sketch of the graph based on these points and properties:
```
y
29 |
27 |------------------------------------------(4,27)
25 |
23 |
21 |
19 |
17 |
15 |
13 |
11 |
9 |
7 | (3,7)
5 |
3 | (2,3)
2 |
1 |--------------------------(1,2.2)-----*
-1 ____________________________________________ x
-1 0 2 3 4 5
(0,2.04)
As you can see, the graph crosses the y-intercept just above 2 and rapidly rises as x increases due to the exponential nature of the function.
### Step 1: Understanding the Function
The given function [tex]\( f(x) = 5^{x-2} + 2 \)[/tex] is an exponential function. The general form of an exponential function is [tex]\( f(x) = a^{b(x-c)} + d \)[/tex], where
- [tex]\( a \)[/tex] is the base of the exponential function (5 in this case),
- [tex]\( b \)[/tex] determines the growth rate,
- [tex]\( c \)[/tex] is the horizontal shift,
- [tex]\( d \)[/tex] is the vertical shift.
For [tex]\( f(x) = 5^{x-2} + 2 \)[/tex]:
- [tex]\( 5 \)[/tex] is the base,
- The exponent [tex]\( x-2 \)[/tex] indicates a horizontal shift of 2 units to the right,
- The constant [tex]\( +2 \)[/tex] represents a vertical shift of 2 units upward.
### Step 2: Identify Key Points
We can find some key points on the graph by plugging in different values of [tex]\( x \)[/tex]:
1. At [tex]\( x = 0 \)[/tex]:
[tex]\[ f(0) = 5^{0-2} + 2 = 5^{-2} + 2 = \frac{1}{25} + 2 \approx 2.04 \][/tex]
2. At [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 5^{2-2} + 2 = 5^0 + 2 = 1 + 2 = 3 \][/tex]
3. At [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 5^{4-2} + 2 = 5^2 + 2 = 25 + 2 = 27 \][/tex]
4. At [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 5^{1-2} + 2 = 5^{-1} + 2 = \frac{1}{5} + 2 = 2.2 \][/tex]
5. At [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 5^{3-2} + 2 = 5^1 + 2 = 5 + 2 = 7 \][/tex]
### Step 3: Sketching the Graph
1. Plot the points computed in Step 2 on a coordinate plane.
- [tex]\( (0, 2.04) \)[/tex]
- [tex]\( (2, 3) \)[/tex]
- [tex]\( (4, 27) \)[/tex]
- [tex]\( (1, 2.2) \)[/tex]
- [tex]\( (3, 7) \)[/tex]
2. Draw the curve through the points. Remember the nature of the exponential function:
- The function approaches [tex]\( y = 2 \)[/tex] (the vertical shift) as [tex]\( x \)[/tex] becomes more negative, but never actually reaches it. This is the horizontal asymptote.
- The function will rise steeply as [tex]\( x \)[/tex] increases since the base of the exponent (5) is greater than 1.
### Step 4: Analyzing the Graph
- Horizontal Asymptote: The function [tex]\( f(x) = 5^{x-2} + 2 \)[/tex] has a horizontal asymptote at [tex]\( y = 2 \)[/tex].
- Behavior:
- For [tex]\( x < 2 \)[/tex], the function value is close to 2.
- For [tex]\( x > 2 \)[/tex], the function value increases rapidly.
### Final Sketch:
Here's a rough sketch of the graph based on these points and properties:
```
y
29 |
27 |------------------------------------------(4,27)
25 |
23 |
21 |
19 |
17 |
15 |
13 |
11 |
9 |
7 | (3,7)
5 |
3 | (2,3)
2 |
1 |--------------------------(1,2.2)-----*
-1 ____________________________________________ x
-1 0 2 3 4 5
(0,2.04)
As you can see, the graph crosses the y-intercept just above 2 and rapidly rises as x increases due to the exponential nature of the function.
Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.
