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### Identifying the Vertex

The vertex form of a quadratic function is [tex]\( f(x) = a(x-h)^2 + k \)[/tex]. What is the vertex of each function? Match the function rule with the coordinates of its vertex.

1. [tex]\( f(x) = 5(x-6)^2 + 9 \)[/tex]
2. [tex]\( f(x) = 9(x-5)^2 + 6 \)[/tex]
3. [tex]\( f(x) = 6(x-5)^2 - 9 \)[/tex]
4. [tex]\( f(x) = 6(x+9)^2 - 5 \)[/tex]
5. [tex]\( f(x) = 9(x+5)^2 - 6 \)[/tex]

Vertices:
- [tex]\( (6, 9) \)[/tex]
- [tex]\( (5, 6) \)[/tex]
- [tex]\( (5, -9) \)[/tex]
- [tex]\( (-9, -5) \)[/tex]
- [tex]\( (-5, -6) \)[/tex]

Sagot :

GinnyAnswer
Let's find the vertex for each given quadratic function by identifying the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] from the standard vertex form, which is [tex]\( f(x) = a(x-h)^2 + k \)[/tex].

### Function 1: [tex]\( f(x)=5(x-6)^2+9 \)[/tex]
Here, the function is already in vertex form:
- [tex]\( a = 5 \)[/tex]
- [tex]\( h = 6 \)[/tex]
- [tex]\( k = 9 \)[/tex]

Thus, the vertex is [tex]\( (6, 9) \)[/tex].

### Function 2: [tex]\( f(x)=9(x-5)^2+6 \)[/tex]
This function also follows the vertex form:
- [tex]\( a = 9 \)[/tex]
- [tex]\( h = 5 \)[/tex]
- [tex]\( k = 6 \)[/tex]

So, the vertex is [tex]\( (5, 6) \)[/tex].

### Function 3: [tex]\( f(x)=6(x-5)^2-9 \)[/tex]
For this function, we have:
- [tex]\( a = 6 \)[/tex]
- [tex]\( h = 5 \)[/tex]
- [tex]\( k = -9 \)[/tex]

The vertex is [tex]\( (5, -9) \)[/tex].

### Function 4: [tex]\( f(x)=6(x+9)^2-5 \)[/tex]
Rewrite [tex]\( x + 9 \)[/tex] as [tex]\( x - (-9) \)[/tex]:
- [tex]\( a = 6 \)[/tex]
- [tex]\( h = -9 \)[/tex]
- [tex]\( k = -5 \)[/tex]

Therefore, the vertex is [tex]\( (-9, -5) \)[/tex].

### Function 5: [tex]\( f(x)=9(x+5)^2-6 \)[/tex]
Rewrite [tex]\( x + 5 \)[/tex] as [tex]\( x - (-5) \)[/tex]:
- [tex]\( a = 9 \)[/tex]
- [tex]\( h = -5 \)[/tex]
- [tex]\( k = -6 \)[/tex]

The vertex is [tex]\( (-5, -6) \)[/tex].

### Summary
Now, we match each function with its vertex coordinate:
1. [tex]\( f(x)=5(x-6)^2+9 \)[/tex] matches to [tex]\( (6, 9) \)[/tex]
2. [tex]\( f(x)=9(x-5)^2+6 \)[/tex] matches to [tex]\( (5, 6) \)[/tex]
3. [tex]\( f(x)=6(x-5)^2-9 \)[/tex] matches to [tex]\( (5, -9) \)[/tex]
4. [tex]\( f(x)=6(x+9)^2-5 \)[/tex] matches to [tex]\( (-9, -5) \)[/tex]
5. [tex]\( f(x)=9(x+5)^2-6 \)[/tex] matches to [tex]\( (-5, -6) \)[/tex]

So, the final matching is:

- [tex]\( f(x)=5(x-6)^2+9 \)[/tex] -> [tex]\( (6, 9) \)[/tex]
- [tex]\( f(x)=9(x-5)^2+6 \)[/tex] -> [tex]\( (5, 6) \)[/tex]
- [tex]\( f(x)=6(x-5)^2-9 \)[/tex] -> [tex]\( (5, -9) \)[/tex]
- [tex]\( f(x)=6(x+9)^2-5 \)[/tex] -> [tex]\( (-9, -5) \)[/tex]
- [tex]\( f(x)=9(x+5)^2-6 \)[/tex] -> [tex]\( (-5, -6) \)[/tex]
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