Graph f(x)=-1/2x^2 if x<1 and 2x+1 if x≥1

To graph the piecewise function \( f(x) \), which is defined as:

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

### Steps to graph:

1. **For \( x < 1 \):**
   - The function is \( f(x) = -\frac{1}{2}x^2 \), which is a downward-facing parabola (since the coefficient of \( x^2 \) is negative). 
   - Since we are only interested in values where \( x < 1 \), we graph this part from \( x = -\infty \) to \( x = 1 \) (not including 1).
   - At \( x = 1 \), we calculate \( f(1) = -\frac{1}{2}(1)^2 = -\frac{1}{2} \). So, the left side of the graph stops just before the point \( (1, -\frac{1}{2}) \).

2. **For \( x \geq 1 \):**
   - The function is \( f(x) = 2x + 1 \), which is a straight line with a slope of 2 and a y-intercept of 1.
   - Starting from \( x = 1 \), the point at \( x = 1 \) is \( f(1) = 2(1) + 1 = 3 \), so the graph starts at \( (1, 3) \) (inclusive) and continues upward for \( x \geq 1 \).

### Key Features:
- The graph is composed of two distinct parts:
  - A downward-facing parabola for \( x < 1 \).
  - A straight line for \( x \geq 1 \).
- At \( x = 1 \), there is a **jump discontinuity** since \( f(x) \) changes from \( -\frac{1}{2} \) (approaching from the left) to 3 (approaching from the right).

Would you like a visual graph or additional details on how to sketch it? 

Here are some related questions:
1. How do you find the domain and range of this piecewise function?
2. What is the derivative of each part of this piecewise function?
3. How does the discontinuity at \( x = 1 \) affect the behavior of the graph?
4. How can you determine if a piecewise function is continuous?
5. What happens if we change the point of discontinuity in this piecewise function?

**Tip:** When dealing with piecewise functions, always pay close attention to how each piece is defined, especially at the boundaries where the pieces meet.

Graph f(x) = -1/2x^2 if x < 1 and 2x + 1 if x ≥ 1

Learn how to graph the piecewise function f(x) = -1/2x^2 for x < 1 and f(x) = 2x + 1 for x ≥ 1. This problem demonstrates how to handle graphing both quadratic and linear functions and examines the discontinuity at x = 1. Suitable for high school students in Grades 10-12.

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