Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
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]
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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Keep exploring Westonci.ca for more insightful answers to your questions. We're here to help.
