At Westonci.ca, we connect you with the answers you need, thanks to our active and informed community. Experience the ease of finding quick and accurate answers to your questions from professionals on our platform. Experience the ease of finding precise answers to your questions from a knowledgeable community of experts.
Sagot :
To determine which of the given functions has a vertex at [tex]\((2, -9)\)[/tex], we'll analyze the vertex form of each function. The vertex form of a quadratic function is given by:
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
Where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's analyze each function individually:
1. Function: [tex]\( f(x) = -(x - 3)^2 \)[/tex]
- This is already in vertex form.
- Here, [tex]\(a = -1\)[/tex], [tex]\(h = 3\)[/tex], [tex]\(k = 0\)[/tex].
- The vertex of this function is [tex]\((3, 0)\)[/tex].
2. Function: [tex]\( f(x) = (x + 8)^2 \)[/tex]
- This is already in vertex form.
- Here, [tex]\(a = 1\)[/tex], [tex]\(h = -8\)[/tex], [tex]\(k = 0\)[/tex].
- The vertex of this function is [tex]\((-8, 0)\)[/tex].
3. Function: [tex]\( f(x) = (x - 5)(x + 1) \)[/tex]
- First, we need to expand this to find the standard form [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ f(x) = x^2 + x - 5x - 5 = x^2 - 4x - 5 \][/tex]
- To find the vertex of this quadratic function, we use the formula for the vertex of a parabola given by a quadratic equation [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\(a = 1\)[/tex] and [tex]\(b = -4\)[/tex]:
[tex]\[ x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2 \][/tex]
- Substitute [tex]\(x = 2\)[/tex] back into the function to find the corresponding [tex]\(y\)[/tex]-coordinate:
[tex]\[ f(2) = (2 - 5)(2 + 1) = (-3)(3) = -9 \][/tex]
- The vertex of this function is [tex]\((2, -9)\)[/tex].
4. Function: [tex]\( f(x) = -(x - 1)(x - 5) \)[/tex]
- First, expand this to find the standard form:
[tex]\[ f(x) = -(x^2 - 6x + 5) = -x^2 + 6x - 5 \][/tex]
- To find the vertex of this quadratic function, we use the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\(a = -1\)[/tex] and [tex]\(b = 6\)[/tex]:
[tex]\[ x = -\frac{6}{2 \times (-1)} = -\frac{6}{-2} = 3 \][/tex]
- Substitute [tex]\(x = 3\)[/tex] back into the function to find the corresponding [tex]\(y\)[/tex]-coordinate:
[tex]\[ f(3) = -(3 - 1)(3 - 5) = -(2)(-2) = 4 \][/tex]
- The vertex of this function is [tex]\((3, 4)\)[/tex].
After analyzing all the functions, the function [tex]\( f(x) = (x - 5)(x + 1) \)[/tex] has the vertex at [tex]\((2, -9)\)[/tex]. Therefore, the correct function is:
[tex]\[ \boxed{f(x) = (x - 5)(x + 1)} \][/tex]
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
Where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
Let's analyze each function individually:
1. Function: [tex]\( f(x) = -(x - 3)^2 \)[/tex]
- This is already in vertex form.
- Here, [tex]\(a = -1\)[/tex], [tex]\(h = 3\)[/tex], [tex]\(k = 0\)[/tex].
- The vertex of this function is [tex]\((3, 0)\)[/tex].
2. Function: [tex]\( f(x) = (x + 8)^2 \)[/tex]
- This is already in vertex form.
- Here, [tex]\(a = 1\)[/tex], [tex]\(h = -8\)[/tex], [tex]\(k = 0\)[/tex].
- The vertex of this function is [tex]\((-8, 0)\)[/tex].
3. Function: [tex]\( f(x) = (x - 5)(x + 1) \)[/tex]
- First, we need to expand this to find the standard form [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ f(x) = x^2 + x - 5x - 5 = x^2 - 4x - 5 \][/tex]
- To find the vertex of this quadratic function, we use the formula for the vertex of a parabola given by a quadratic equation [tex]\(ax^2 + bx + c\)[/tex]:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\(a = 1\)[/tex] and [tex]\(b = -4\)[/tex]:
[tex]\[ x = -\frac{-4}{2 \times 1} = \frac{4}{2} = 2 \][/tex]
- Substitute [tex]\(x = 2\)[/tex] back into the function to find the corresponding [tex]\(y\)[/tex]-coordinate:
[tex]\[ f(2) = (2 - 5)(2 + 1) = (-3)(3) = -9 \][/tex]
- The vertex of this function is [tex]\((2, -9)\)[/tex].
4. Function: [tex]\( f(x) = -(x - 1)(x - 5) \)[/tex]
- First, expand this to find the standard form:
[tex]\[ f(x) = -(x^2 - 6x + 5) = -x^2 + 6x - 5 \][/tex]
- To find the vertex of this quadratic function, we use the vertex formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Here, [tex]\(a = -1\)[/tex] and [tex]\(b = 6\)[/tex]:
[tex]\[ x = -\frac{6}{2 \times (-1)} = -\frac{6}{-2} = 3 \][/tex]
- Substitute [tex]\(x = 3\)[/tex] back into the function to find the corresponding [tex]\(y\)[/tex]-coordinate:
[tex]\[ f(3) = -(3 - 1)(3 - 5) = -(2)(-2) = 4 \][/tex]
- The vertex of this function is [tex]\((3, 4)\)[/tex].
After analyzing all the functions, the function [tex]\( f(x) = (x - 5)(x + 1) \)[/tex] has the vertex at [tex]\((2, -9)\)[/tex]. Therefore, the correct function is:
[tex]\[ \boxed{f(x) = (x - 5)(x + 1)} \][/tex]
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.