Sure, let's analyze the given problem step-by-step to determine which function represents the specified quadratic.
### Information Given:
1. Vertex of the Parabola:
The minimum of the quadratic function is at [tex]\((-1, 2)\)[/tex].
This implies:
- The vertex form of a quadratic function is [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
- Here, [tex]\( h = -1 \)[/tex] and [tex]\( k = 2 \)[/tex]. So, [tex]\( f(x) = a(x + 1)^2 + 2 \)[/tex].
2. Point on the Parabola:
The point [tex]\((2, 20)\)[/tex] lies on the parabola.
We need to check each given function to see if it passes through the point [tex]\((2, 20)\)[/tex].
### Testing the Functions:
1. Function: [tex]\( f(x) = (x + 1)^2 + 2 \)[/tex]
```
f(2) = (2 + 1)^2 + 2
= 3^2 + 2
= 9 + 2
= 11
```
The value is 11, which does not equal 20. So, this is not the correct function.
2. Function: [tex]\( f(x) = (x - 1)^2 + 2 \)[/tex]
```
f(2) = (2 - 1)^2 + 2
= 1^2 + 2
= 1 + 2
= 3
```
The value is 3, which does not equal 20. So, this is not the correct function.
3. Function: [tex]\( f(x) = 2(x + 1)^2 + 2 \)[/tex]
```
f(2) = 2(2 + 1)^2 + 2
= 2(3)^2 + 2
= 2(9) + 2
= 18 + 2
= 20
```
The value is 20, which matches the given point. So, this is likely the correct function.
4. Function: [tex]\( f(x) = 2(x - 1)^2 + 2 \)[/tex]
```
f(2) = 2(2 - 1)^2 + 2
= 2(1)^2 + 2
= 2(1) + 2
= 2 + 2
= 4
```
The value is 4, which does not equal 20. So, this is not the correct function.
### Conclusion:
The function which correctly represents the situation where the minimum of the graph is at [tex]\((-1, 2)\)[/tex] and the point [tex]\((2, 20)\)[/tex] is on the parabola is:
[tex]\[ f(x) = 2(x + 1)^2 + 2 \][/tex]
