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create 1 piecewise graph for each problem

Here are the graphs for the three piecewise-defined functions:

1. **Function 1**:
   \[
   y = \begin{cases} 
   -x^2, & x \leq 1 \\
   x - 1, & x > 1 
   \end{cases}
   \]
   This graph shows a parabolic curve for \(x \leq 1\) and a linear function for \(x > 1\).

2. **Function 2**:
   \[
   y = \begin{cases} 
   x + 9, & x < -3 \\
   -2x, & -3 \leq x \leq 3 \\
   -6, & x > 3 
   \end{cases}
   \]
   This graph is composed of a linear function with a slope of 1 for \(x < -3\), a different linear function with a slope of -2 for \(-3 \leq x \leq 3\), and a constant function for \(x > 3\).

3. **Function 3**:
   \[
   y = \begin{cases} 
   |3 + x|, & x < -1 \\
   (x - 1)^2, & x \geq -1 
   \end{cases}
   \]
   This graph features an absolute value function for \(x < -1\) and a parabola for \(x \geq -1\).

Would you like further clarification on any of these graphs?

Here are some related questions to further your understanding:
1. How can you determine the continuity of a piecewise function?
2. What effect does a parabola's vertex have on the shape of a graph?
3. How can you calculate the slope of a linear piecewise segment?
4. In what situations is an absolute value function used in real-life scenarios?
5. How do discontinuities in piecewise functions appear graphically?

**Tip**: When graphing piecewise functions, pay attention to whether each segment is open or closed at the boundary points to ensure the graph is accurate.

Sketch the graph of the following piecewise-defined functions:
1. y = { -x^2, x ≤ 1; x - 1, x > 1 }
2. y = { x + 9, x < -3; -2x, -3 ≤ x ≤ 3; -6, x > 3 }
3. y = { |3 + x|, x < -1; (x - 1)^2, x ≥ -1 }

This page provides a step-by-step guide on graphing piecewise-defined functions involving quadratic, linear, and absolute value segments. Learn how to break down and graph each piece, paying attention to boundary points and continuity.

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