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:
- Vertex: The vertex \((h, k)\) is the point where the parabola changes direction.
- 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.
- 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:
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Identify the Vertex: Locate the vertex of the parabola from the graph. This gives you \(h\) and \(k\).
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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 \(a\): \[ a = \frac{y_1 - k}{(x_1 - h)^2} \]
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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:
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Vertex: \(h = 2\), \(k = 3\).
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Using another point: With \((4, 7)\): \[ 7 = a(4 - 2)^2 + 3 \] Simplifying gives: \[ 7 = a(2)^2 + 3 \] \[ 7 = 4a + 3 \] \[ 4 = 4a \implies a = 1 \]
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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.
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