Answer
To sketch the graph of the function:
\[
g(x) = \begin{cases}
x + 2, & \text{if } x \leq -4 \\
\frac{1}{2}x - 6, & \text{if } x > -4
\end{cases}
\]
1. **Plot the first piece (\( x + 2 \))**:
- At \( x = -4 \), \( g(-4) = -2 \). Plot a closed circle at \((-4, -2)\).
- Draw a straight line with a slope of 1 extending to the left.
2. **Plot the second piece (\( \frac{1}{2}x - 6 \))**:
- At \( x = -4 \), \( g(-4) = -8 \). Plot an open circle at \((-4, -8)\).
- Draw a straight line with a slope of \( \frac{1}{2} \) extending to the right.
3. **Indicate the jump discontinuity** at \( x = -4 \) with the closed and open circles.
The graph consists of two straight lines meeting at \( x = -4 \) with a jump, showing the piecewise nature of the function.
Solution
To sketch the graph of the piecewise function:
\[
g(x) = \begin{cases}
x + 2, & \text{if } x \leq -4 \\
\frac{1}{2}x - 6, & \text{if } x > -4
\end{cases}
\]
follow these steps:
### 1. **Identify the Pieces of the Function**
- **First Piece:** \( g(x) = x + 2 \) for \( x \leq -4 \)
- **Slope:** 1
- **Y-intercept:** 2
- **Second Piece:** \( g(x) = \frac{1}{2}x - 6 \) for \( x > -4 \)
- **Slope:** \( \frac{1}{2} \)
- **Y-intercept:** -6
### 2. **Determine Key Points and Continuity at \( x = -4 \)**
- **At \( x = -4 \):**
- Use the first piece since \( x \leq -4 \).
- \( g(-4) = (-4) + 2 = -2 \)
- This point is included in the first piece: **\((-4, -2)\)** (closed circle).
- **Limit from the Right as \( x \) Approaches -4:**
- Use the second piece: \( \lim_{x \to -4^+} g(x) = \frac{1}{2}(-4) - 6 = -2 - 6 = -8 \)
- This indicates a **jump discontinuity** at \( x = -4 \).
- Represented by an open circle at \((-4, -8)\).
### 3. **Plot Each Piece**
- **First Piece (\( g(x) = x + 2 \) for \( x \leq -4 \)):**
- Plot the point \((-4, -2)\) with a closed circle.
- Since the slope is 1, for every 1 unit increase in \( x \), \( y \) increases by 1.
- Draw a straight line extending to the left from \((-4, -2)\).
- **Second Piece (\( g(x) = \frac{1}{2}x - 6 \) for \( x > -4 \)):**
- Start just to the right of \( x = -4 \) with an open circle at \((-4, -8)\).
- Since the slope is \( \frac{1}{2} \), for every 2 units increase in \( x \), \( y \) increases by 1.
- Draw a straight line extending to the right from the open circle.
### 4. **Visual Representation**
Here's a rough sketch of the graph:
```
y
|
| Second Piece: y = (1/2)x - 6
| /
| /
| /
| /
|__________●________/__________ x
(-4, -2) Open Circle at (-4, -8)
|
|
| First Piece: y = x + 2
|
|
|
|
```
### 5. **Key Characteristics**
- **Slope:** The first piece has a steeper slope (1) compared to the second piece (\( \frac{1}{2} \)).
- **Y-Intercepts:**
- First piece intersects the y-axis at (0, 2).
- Second piece intersects the y-axis at (0, -6).
- **Discontinuity:** There is a jump at \( x = -4 \):
- The function value jumps from \(-2\) (from the first piece) to approaching \(-8\) (from the second piece) as \( x \) crosses \(-4\) from left to right.
### 6. **Summary**
- For \( x \leq -4 \), the graph is a straight line with slope 1 passing through \((-4, -2)\).
- For \( x > -4 \), the graph is a different straight line with slope \( \frac{1}{2} \) starting just above \((-4, -8)\).
- There's a visible jump at \( x = -4 \) indicating the discontinuity between the two pieces.
By following these steps, you can accurately sketch the graph of the given piecewise function.
Reviewed and approved by the UpStudy tutoring team
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