To determine the equation of the quadratic function in vertex form, we need to identify the vertex of the graph in question and then see how the provided options match that vertex.
The vertex form of a quadratic function is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola.
Given the answer choices:
- \(-\frac{1}{2} (x + 3)^2 + 2\)
- \(-\frac{1}{2} (x - 3)^2 + 2\)
- \(\frac{1}{2} (x - 3)^2 + 2\)
- \(-\frac{1}{2} (x + 3)^2 - 2\)
Step 1: Identify the Vertex
You need to identify the vertex from the graph. The vertex can be determined by looking for the point where the parabola reaches its maximum or minimum value (the highest or lowest point).
Step 2: Match with Options
Assuming you found the vertex to be \((-3, 2)\), the vertex form would be:
\[ f(x) = a(x + 3)^2 + 2 \]
For a parabola opening downwards, \(a\) will be negative (less than 0), and for upward opening, \(a\) will be positive.
If the vertex was found at \((3, 2)\), then in vertex form, it could be:
\[ f(x) = a(x - 3)^2 + 2 \]
Step 3: Analyze Each Option
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Option 1: \(-\frac{1}{2} (x + 3)^2 + 2\)
- Vertex: \((-3, 2)\), and opens downwards.
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Option 2: \(-\frac{1}{2} (x - 3)^2 + 2\)
- Vertex: \((3, 2)\), and opens downwards.
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Option 3: \(\frac{1}{2} (x - 3)^2 + 2\)
- Vertex: \((3, 2)\), and opens upwards.
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Option 4: \(-\frac{1}{2} (x + 3)^2 - 2\)
- Vertex: \((-3, -2)\), and opens downwards.
Conclusion
You would pick the option that correctly states the location of the vertex based on your graph. If the vertex you found is \((-3, 2)\) and the parabola opens downwards, then Option 1 \(-\frac{1}{2} (x + 3)^2 + 2\) is the correct answer.
If you provide the details of the graph (like the coordinates of the vertex and its direction), I can confirm the best match from the options.
2 answers