To write the piecewise function based on the description of the graph, we analyze each part of the graph and its corresponding mathematical representation:
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First segment:
- A straight line that starts in quadrant 3 with a positive slope and has a solid endpoint at \((-1, -1)\).
- Since it is given that the endpoint is solid, the function value at \(x = -1\) would be included.
- The line can be calculated by finding its slope, which appears to be \(\frac{3}{2}\) (based on adjusting for the rise and run).
- Thus, the equation for this line is \(f(x) = \frac{3}{2}x + \frac{1}{2}\) for \(x \leq -1\).
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Second segment:
- A quadratic function opening upwards from an open circle at \((-1, -2)\) to a closed circle at \((3, 2)\).
- The vertex form of a quadratic function can be expressed as \(f(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex.
- This equation has been identified as \(f(x) = \frac{1}{4}(x + 1)^2 - 2\) for \(-1 < x \leq 3\).
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Third segment:
- A quadratic function opening downwards with a vertex at \((5, 4)\) starting from the closed endpoint at \((3, 2)\) to another closed endpoint at \((9, -4)\).
- The vertex form is \(f(x) = -\frac{1}{2}(x - 5)^2 + 4\) for \(3 < x \leq 9\).
Now we can summarize this in a piecewise function as follows:
\[ f(x) = \begin{cases} \frac{3}{2}x + \frac{1}{2}, & x \leq -1 \ \frac{1}{4}(x + 1)^2 - 2, & -1 < x \leq 3 \ -\frac{1}{2}(x - 5)^2 + 4, & 3 < x \leq 9 \end{cases} \]
This is the mathematical representation of the piecewise function based on the given description of the graph.