Westonci.ca makes finding answers easy, with a community of experts ready to provide you with the information you seek. Get quick and reliable solutions to your questions from a community of experienced experts on our platform. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
To find the equation of the quadratic function in both vertex and standard form, given the vertex and a point it passes through, we follow these steps:
1. Vertex Form of the Quadratic Function:
The vertex form of a quadratic function is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola. Given the vertex [tex]\((-1, 6)\)[/tex], we can substitute [tex]\(h = -1\)[/tex] and [tex]\(k = 6\)[/tex] into the vertex form:
[tex]\[ y = a(x - (-1))^2 + 6 \][/tex]
Simplifying the expression inside the parentheses, we get:
[tex]\[ y = a(x + 1)^2 + 6 \][/tex]
2. Determine the Value of [tex]\(a\)[/tex]:
We are given a point [tex]\((2, 24)\)[/tex] that the function passes through. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 24\)[/tex] into the equation:
[tex]\[ 24 = a(2 + 1)^2 + 6 \][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[ 24 = a(3)^2 + 6 \][/tex]
[tex]\[ 24 = 9a + 6 \][/tex]
Subtract 6 from both sides:
[tex]\[ 18 = 9a \][/tex]
Divide by 9:
[tex]\[ a = 2 \][/tex]
3. Write the Vertex Form:
Substitute [tex]\(a = 2\)[/tex] back into the vertex form:
[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]
4. Convert to Standard Form:
To find the standard form [tex]\(y = ax^2 + bx + c\)[/tex], expand the vertex form:
[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]
First, expand [tex]\((x + 1)^2\)[/tex]:
[tex]\[ (x + 1)^2 = x^2 + 2x + 1 \][/tex]
Substitute back into the equation:
[tex]\[ y = 2(x^2 + 2x + 1) + 6 \][/tex]
Distribute the 2:
[tex]\[ y = 2x^2 + 4x + 2 + 6 \][/tex]
Combine like terms:
[tex]\[ y = 2x^2 + 4x + 8 \][/tex]
Thus, the equations of the quadratic function are:
- Vertex form: [tex]\(\boxed{y = 2(x + 1)^2 + 6}\)[/tex]
- Standard form: [tex]\(\boxed{y = 2x^2 + 4x + 8}\)[/tex]
1. Vertex Form of the Quadratic Function:
The vertex form of a quadratic function is given by:
[tex]\[ y = a(x - h)^2 + k \][/tex]
Here, [tex]\((h, k)\)[/tex] is the vertex of the parabola. Given the vertex [tex]\((-1, 6)\)[/tex], we can substitute [tex]\(h = -1\)[/tex] and [tex]\(k = 6\)[/tex] into the vertex form:
[tex]\[ y = a(x - (-1))^2 + 6 \][/tex]
Simplifying the expression inside the parentheses, we get:
[tex]\[ y = a(x + 1)^2 + 6 \][/tex]
2. Determine the Value of [tex]\(a\)[/tex]:
We are given a point [tex]\((2, 24)\)[/tex] that the function passes through. Substitute [tex]\(x = 2\)[/tex] and [tex]\(y = 24\)[/tex] into the equation:
[tex]\[ 24 = a(2 + 1)^2 + 6 \][/tex]
Solve for [tex]\(a\)[/tex]:
[tex]\[ 24 = a(3)^2 + 6 \][/tex]
[tex]\[ 24 = 9a + 6 \][/tex]
Subtract 6 from both sides:
[tex]\[ 18 = 9a \][/tex]
Divide by 9:
[tex]\[ a = 2 \][/tex]
3. Write the Vertex Form:
Substitute [tex]\(a = 2\)[/tex] back into the vertex form:
[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]
4. Convert to Standard Form:
To find the standard form [tex]\(y = ax^2 + bx + c\)[/tex], expand the vertex form:
[tex]\[ y = 2(x + 1)^2 + 6 \][/tex]
First, expand [tex]\((x + 1)^2\)[/tex]:
[tex]\[ (x + 1)^2 = x^2 + 2x + 1 \][/tex]
Substitute back into the equation:
[tex]\[ y = 2(x^2 + 2x + 1) + 6 \][/tex]
Distribute the 2:
[tex]\[ y = 2x^2 + 4x + 2 + 6 \][/tex]
Combine like terms:
[tex]\[ y = 2x^2 + 4x + 8 \][/tex]
Thus, the equations of the quadratic function are:
- Vertex form: [tex]\(\boxed{y = 2(x + 1)^2 + 6}\)[/tex]
- Standard form: [tex]\(\boxed{y = 2x^2 + 4x + 8}\)[/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.
