Question
Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.
Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.
Given the graph of a quadratic​ function, write the equation of the function in vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k.
Answers
GPT-4o mini
To write the equation of a quadratic function in vertex form \( f(x) = a(x - h)^2 + k \), you need to identify several key features from the graph:
1. **Vertex**: The vertex \((h, k)\) is the point where the parabola changes direction.
2. **Direction**: The value of \(a\) determines the direction of the opening of the parabola:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
3. **Width**: The absolute value of \(a\) affects the width of the parabola. A larger value of \(|a|\) makes the parabola narrower, while a smaller value makes it wider.
### Steps to write the equation:
1. **Identify the Vertex**: Locate the vertex of the parabola from the graph. This gives you \(h\) and \(k\).
2. **Determine the Value of \(a\)**:
- Choose another point on the parabola (not the vertex), and note its coordinates \((x_1, y_1)\).
- Use the vertex and this point to find \(a\) using the formula:
\[
y_1 = a(x_1 - h)^2 + k
\]
- Rearrange to solve for \(a\):
\[
a = \frac{y_1 - k}{(x_1 - h)^2}
\]
3. **Write the Equation**: Plug the values of \(a\), \(h\), and \(k\) back into the vertex form equation:
\[
f(x) = a(x - h)^2 + k
\]
### Example:
Suppose we identified the vertex as \((2, 3)\) and another point on the graph as \((4, 7)\). Here's how you would determine the equation:
1. **Vertex**: \(h = 2\), \(k = 3\).
2. **Using another point**: With \((4, 7)\):
\[
7 = a(4 - 2)^2 + 3
\]
Simplifying gives:
\[
7 = a(2)^2 + 3
\]
\[
7 = 4a + 3
\]
\[
4 = 4a \implies a = 1
\]
3. **Equation of the Function**:
\[
f(x) = 1(x - 2)^2 + 3 \implies f(x) = (x - 2)^2 + 3
\]
You would follow these steps using the specific details from the graph of the quadratic function you have.
1. **Vertex**: The vertex \((h, k)\) is the point where the parabola changes direction.
2. **Direction**: The value of \(a\) determines the direction of the opening of the parabola:
- If \(a > 0\), the parabola opens upwards.
- If \(a < 0\), the parabola opens downwards.
3. **Width**: The absolute value of \(a\) affects the width of the parabola. A larger value of \(|a|\) makes the parabola narrower, while a smaller value makes it wider.
### Steps to write the equation:
1. **Identify the Vertex**: Locate the vertex of the parabola from the graph. This gives you \(h\) and \(k\).
2. **Determine the Value of \(a\)**:
- Choose another point on the parabola (not the vertex), and note its coordinates \((x_1, y_1)\).
- Use the vertex and this point to find \(a\) using the formula:
\[
y_1 = a(x_1 - h)^2 + k
\]
- Rearrange to solve for \(a\):
\[
a = \frac{y_1 - k}{(x_1 - h)^2}
\]
3. **Write the Equation**: Plug the values of \(a\), \(h\), and \(k\) back into the vertex form equation:
\[
f(x) = a(x - h)^2 + k
\]
### Example:
Suppose we identified the vertex as \((2, 3)\) and another point on the graph as \((4, 7)\). Here's how you would determine the equation:
1. **Vertex**: \(h = 2\), \(k = 3\).
2. **Using another point**: With \((4, 7)\):
\[
7 = a(4 - 2)^2 + 3
\]
Simplifying gives:
\[
7 = a(2)^2 + 3
\]
\[
7 = 4a + 3
\]
\[
4 = 4a \implies a = 1
\]
3. **Equation of the Function**:
\[
f(x) = 1(x - 2)^2 + 3 \implies f(x) = (x - 2)^2 + 3
\]
You would follow these steps using the specific details from the graph of the quadratic function you have.