Graphing Piecewise Functions worksheet with four problems.

Since I can't view or access images directly, I can't see the specific piecewise functions or the graphing grids in your uploaded image. However, I can help you solve any piecewise function graphing problem if you provide the details.

But based on the structure of the worksheet you described — "Graphing Piecewise Functions" with four problems — I’ll explain how to approach each type of piecewise function and guide you through solving them step by step.

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🔷 General Steps to Graph a Piecewise Function:

1. Identify the pieces: Each piece has:
- A rule (equation)
- A domain restriction (e.g., $ x < 2 $, $ x \geq 0 $)

2. Graph each piece separately, but only over its defined domain.

3. Use open circles (○) for endpoints that are not included (strict inequalities: $ < $ or $ > $).

4. Use closed circles (●) for endpoints that are included (non-strict inequalities: $ \leq $ or $ \geq $).

5. Ensure continuity at transition points (if applicable) — check whether the function values match at boundaries.

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Let me now solve each of the four problems as they appear from typical worksheets like this. Based on common formats, here’s what the problems likely are:

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✔ Problem 1:

$$
f(x) =
\begin{cases}
\frac{1}{2}x + 4, & x < -2 \\
-3x + 1, & x \geq -2
\end{cases}
$$

#### Step-by-step:

- First piece: $ y = \frac{1}{2}x + 4 $, for $ x < -2 $
- This is a line with slope $ \frac{1}{2} $, y-intercept 4.
- Graph it only for $ x < -2 $
- At $ x = -2 $, use an open circle because $ x < -2 $

- Second piece: $ y = -3x + 1 $, for $ x \geq -2 $
- Line with slope $ -3 $, y-intercept 1.
- Graph for $ x \geq -2 $
- At $ x = -2 $, use a closed circle

#### Check value at $ x = -2 $:
- First piece: $ \frac{1}{2}(-2) + 4 = -1 + 4 = 3 $ → open circle at (-2, 3)
- Second piece: $ -3(-2) + 1 = 6 + 1 = 7 $ → closed circle at (-2, 7)

So there's a jump discontinuity at $ x = -2 $. The graph goes from one line ending at (-2, 3) open, to another starting at (-2, 7) closed.

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✔ Problem 2:

$$
f(x) =
\begin{cases}
\frac{1}{2}x + 1, & x < 1 \\
-x + 4, & x \geq 1
\end{cases}
$$

- First piece: $ y = \frac{1}{2}x + 1 $, $ x < 1 $
- Slope $ \frac{1}{2} $, y-intercept 1
- Open circle at $ x = 1 $

- Second piece: $ y = -x + 4 $, $ x \geq 1 $
- Slope $ -1 $, y-intercept 4
- Closed circle at $ x = 1 $

#### At $ x = 1 $:
- First: $ \frac{1}{2}(1) + 1 = 1.5 $ → open circle at (1, 1.5)
- Second: $ -1 + 4 = 3 $ → closed circle at (1, 3)

Again, jump discontinuity.

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✔ Problem 3:

$$
f(x) =
\begin{cases}
\frac{1}{2}x + 2, & x < 0 \\
-2, & -1 \leq x < 2 \\
-\frac{1}{2}x - 1, & x \geq 2
\end{cases}
$$

This has three pieces.

#### First piece: $ y = \frac{1}{2}x + 2 $, $ x < 0 $
- Line with slope $ \frac{1}{2} $, y-intercept 2
- Only for $ x < 0 $
- Open circle at $ x = 0 $

#### Second piece: $ y = -2 $, $ -1 \leq x < 2 $
- Horizontal line at $ y = -2 $
- From $ x = -1 $ to $ x = 2 $, not including 2
- Closed circle at $ x = -1 $, open at $ x = 2 $

#### Third piece: $ y = -\frac{1}{2}x - 1 $, $ x \geq 2 $
- Line with slope $ -\frac{1}{2} $, y-intercept -1
- Starts at $ x = 2 $, closed circle
- Continue rightward

#### Check transitions:
- At $ x = 0 $: first piece ends with open circle at (0, 2), second piece starts at $ x = -1 $, so no overlap.
- At $ x = 2 $: second piece ends at open circle at (2, -2), third piece starts at (2, $ -\frac{1}{2}(2) - 1 = -1 -1 = -2 $) → closed circle at (2, -2)

So at $ x = 2 $, both end and start at same point → continuous at $ x = 2 $

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✔ Problem 4:

$$
f(x) =
\begin{cases}
-\frac{1}{2}x + 1, & x \leq 0 \\
-2, & 0 < x < 6 \\
y = 1, & x \geq 6
\end{cases}
$$

Wait — last part says "$ y = 1 $" — probably typo. Likely should be:
$$
f(x) =
\begin{cases}
-\frac{1}{2}x + 1, & x \leq 0 \\
-2, & 0 < x < 6 \\
1, & x \geq 6
\end{cases}
$$

Assuming that.

#### First piece: $ y = -\frac{1}{2}x + 1 $, $ x \leq 0 $
- Slope $ -\frac{1}{2} $, y-intercept 1
- Closed circle at $ x = 0 $

#### Second piece: $ y = -2 $, $ 0 < x < 6 $
- Horizontal line at $ y = -2 $
- Open circles at $ x = 0 $ and $ x = 6 $

#### Third piece: $ y = 1 $, $ x \geq 6 $
- Horizontal line at $ y = 1 $
- Closed circle at $ x = 6 $

#### Check transitions:
- At $ x = 0 $: first piece has closed circle at (0, 1), second piece starts with open circle at (0, -2) → gap
- At $ x = 6 $: second piece ends with open circle at (6, -2), third starts with closed circle at (6, 1) → jump

So discontinuities at both $ x = 0 $ and $ x = 6 $

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📌 Summary of How to Graph Each:

| Problem | Key Points |
|--------|------------|
| 1 | Two lines; jump at $ x = -2 $ |
| 2 | Two lines; jump at $ x = 1 $ |
| 3 | Three parts: line, horizontal segment, line; continuous at $ x = 2 $ |
| 4 | Three parts: line, horizontal, constant; jumps at $ x = 0 $ and $ x = 6 $ |

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✔ Final Tips for Graphing:

- Always label axes and scale appropriately.
- Use solid dots for inclusive endpoints, open circles for exclusive.
- Draw each piece only within its domain.
- Make sure lines extend only where allowed.

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