https://file.aiquickdraw.com/mathbot/file/1726432803368c9iae93v.jpg

The piecewise function provided is:

\[
f(x) =
\begin{cases} 
\frac{1}{2}x^2 - 1 & \text{if } x \leq -2 \\
-\frac{1}{2}x + 5 & \text{if } x > -2 
\end{cases}
\]

To graph this function, follow these steps:

### 1. For \(x \leq -2\), the function is quadratic:
\[
f(x) = \frac{1}{2}x^2 - 1
\]
This is a parabola opening upwards because the coefficient of \(x^2\) is positive (\(\frac{1}{2}\)). The vertex of the parabola occurs at \(x = 0\), but we are only interested in the portion for \(x \leq -2\).

### 2. For \(x > -2\), the function is linear:
\[
f(x) = -\frac{1}{2}x + 5
\]
This is a straight line with a slope of \(-\frac{1}{2}\) and a y-intercept at 5.

### 3. Behavior at \(x = -2\):
At \(x = -2\), check the function values from both pieces:
- From the quadratic part: \(f(-2) = \frac{1}{2}(-2)^2 - 1 = 2 - 1 = 1\).
- From the linear part: \(f(-2)\) is undefined in the linear part since it only applies for \(x > -2\).

Thus, the function is continuous at \(x = -2\) with \(f(-2) = 1\).

### Graph:

- For \(x \leq -2\), plot the parabola \(\frac{1}{2}x^2 - 1\) starting at \(x = -2\) (where \(f(-2) = 1\)).
- For \(x > -2\), plot the line \(-\frac{1}{2}x + 5\).

Let me know if you need a detailed graph, and I can generate it for you.

Would you like further details or have any specific questions?

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### Related Questions:
1. How do you determine the vertex of a quadratic function?
2. What does it mean for a function to be continuous at a point?
3. How do you find the y-intercept of a linear function?
4. Can you explain the process of graphing a piecewise function in more detail?
5. How can you determine if a piecewise function is differentiable at a point?

### Tip:
When graphing piecewise functions, always check for continuity at the boundary points where the function changes its definition.

Graph the piecewise function: f(x) = (1/2)x^2 - 1 for x ≤ -2 and f(x) = -(1/2)x + 5 for x > -2.

Learn how to graph a piecewise function that combines quadratic and linear parts. This problem covers key concepts such as piecewise functions, continuity, and the behavior of functions at boundary points. Suitable for high school students.

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