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Sagot :
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]
### 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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