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.
