To determine the vertex of each quadratic function given in the vertex form [tex]\( f(x) = a(x-h)^2 + k \)[/tex], we need to identify the values of [tex]\( h \)[/tex] and [tex]\( k \)[/tex] for each function. The vertex of the function is then given by the coordinates [tex]\( (h, k) \)[/tex].
Let's analyze each function in the vertex form one by one.
1. Function: [tex]\( f(x) = 9(x-5)^2 + 6 \)[/tex]
- Here, [tex]\( a = 9 \)[/tex], [tex]\( h = 5 \)[/tex], and [tex]\( k = 6 \)[/tex].
- Therefore, the vertex of the function is [tex]\((5, 6)\)[/tex].
2. Function: [tex]\( f(x) = 6(x+9)^2 - 5 \)[/tex]
- Here, [tex]\( a = 6 \)[/tex], [tex]\( h = -9 \)[/tex] (note the sign change due to [tex]\(+9\)[/tex] being [tex]\((x - (-9))\)[/tex]), and [tex]\( k = -5 \)[/tex].
- Therefore, the vertex of the function is [tex]\((-9, -5)\)[/tex].
3. Function: [tex]\( f(x) = 6(x-5)^2 - 9 \)[/tex]
- Here, [tex]\( a = 6 \)[/tex], [tex]\( h = 5 \)[/tex], and [tex]\( k = -9 \)[/tex].
- Therefore, the vertex of the function is [tex]\((5, -9)\)[/tex].
4. Function: [tex]\( f(x) = 5(x-6)^2 + 9 \)[/tex]
- Here, [tex]\( a = 5 \)[/tex], [tex]\( h = 6 \)[/tex], and [tex]\( k = 9 \)[/tex].
- Therefore, the vertex of the function is [tex]\((6, 9)\)[/tex].
5. Function: [tex]\( f(x) = 9(x+5)^2 - 6 \)[/tex]
- Here, [tex]\( a = 9 \)[/tex], [tex]\( h = -5 \)[/tex] (note the sign change due to [tex]\(+5\)[/tex] being [tex]\((x - (-5))\)[/tex]), and [tex]\( k = -6 \)[/tex].
- Therefore, the vertex of the function is [tex]\((-5, -6)\)[/tex].
Now, let's match each function to the coordinates of its vertex:
- [tex]\( f(x) = 9(x-5)^2 + 6 \)[/tex] has the vertex [tex]\((5, 6)\)[/tex].
- [tex]\( f(x) = 6(x+9)^2 - 5 \)[/tex] has the vertex [tex]\((-9, -5)\)[/tex].
- [tex]\( f(x) = 6(x-5)^2 - 9 \)[/tex] has the vertex [tex]\((5, -9)\)[/tex].
- [tex]\( f(x) = 5(x-6)^2 + 9 \)[/tex] has the vertex [tex]\((6, 9)\)[/tex].
- [tex]\( f(x) = 9(x+5)^2 - 6 \)[/tex] has the vertex [tex]\((-5, -6)\)[/tex].
Thus, the matches are:
1. [tex]\( (5, 6) \)[/tex] - [tex]\( f(x) = 9(x-5)^2 + 6 \)[/tex]
2. [tex]\( (-9, -5) \)[/tex] - [tex]\( f(x) = 6(x+9)^2 - 5 \)[/tex]
3. [tex]\( (5, -9) \)[/tex] - [tex]\( f(x) = 6(x-5)^2 - 9 \)[/tex]
4. [tex]\( (6, 9) \)[/tex] - [tex]\( f(x) = 5(x-6)^2 + 9 \)[/tex]
5. [tex]\( (-5, -6) \)[/tex] - [tex]\( f(x) = 9(x+5)^2 - 6 \)[/tex]
