Algebra 2 YL 4.4 Graphing Piecewise Functions Worksheet with six piecewise function problems and blank graphs for plotting.

Let's solve each of the piecewise functions and explain how to graph them step by step. I'll go through each problem, describe the function, determine the correct graphing method, and provide a clear explanation.

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1.

$$
f(x) =
\begin{cases}
-x & \text{if } x \leq 2 \\
x & \text{if } x > 2
\end{cases}
$$

#### Step-by-step:
- For $ x \leq 2 $: Graph $ y = -x $. This is a line with slope $-1$, passing through points like $ (0,0), (2,-2), (-2,2) $.
- Since $ x \leq 2 $, include the point at $ x = 2 $: $ f(2) = -2 $. Use a closed circle at $ (2, -2) $.
- For $ x > 2 $: Graph $ y = x $. This is a line with slope $1$, starting just after $ x = 2 $.
- At $ x = 2 $, $ f(x) = x $ would be $ 2 $, but since $ x > 2 $, we do not include $ x = 2 $ here.
- Use an open circle at $ (2, 2) $, then draw the line $ y = x $ for $ x > 2 $.

#### Graph Description:
- Left side: Line $ y = -x $ from left up to $ (2, -2) $ (closed).
- Right side: Line $ y = x $ starting from $ (2, 2) $ (open) going upward.

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2.

$$
f(x) =
\begin{cases}
2 & \text{if } x > -3 \\
-5 & \text{if } x < -3
\end{cases}
$$

Wait — this function has no definition at $ x = -3 $. Let’s check:

- $ x > -3 $ → $ f(x) = 2 $
- $ x < -3 $ → $ f(x) = -5 $

But nothing is defined at $ x = -3 $. That’s okay — we can still graph it.

#### Step-by-step:
- For $ x < -3 $: Constant function $ f(x) = -5 $. Draw a horizontal line at $ y = -5 $ for all $ x < -3 $.
- Use an open circle at $ x = -3 $ because $ x < -3 $ does not include $ x = -3 $.
- For $ x > -3 $: Constant function $ f(x) = 2 $. Draw a horizontal line at $ y = 2 $ for all $ x > -3 $.
- Use an open circle at $ x = -3 $ here too, since $ x > -3 $ doesn't include $ x = -3 $.

At $ x = -3 $, there is a gap — no value defined.

#### Graph Description:
- Two horizontal lines:
- $ y = -5 $ for $ x < -3 $, open circle at $ (-3, -5) $
- $ y = 2 $ for $ x > -3 $, open circle at $ (-3, 2) $
- No point at $ x = -3 $

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3.

$$
f(x) =
\begin{cases}
-1 & \text{if } x \leq -2 \\
2 & \text{if } x > -2
\end{cases}
$$

#### Step-by-step:
- For $ x \leq -2 $: $ f(x) = -1 $. Horizontal line at $ y = -1 $, including $ x = -2 $ → closed circle at $ (-2, -1) $.
- For $ x > -2 $: $ f(x) = 2 $. Horizontal line at $ y = 2 $, starting just after $ x = -2 $ → open circle at $ (-2, 2) $.

#### Graph Description:
- Left: Horizontal line $ y = -1 $ for $ x \leq -2 $, closed at $ (-2, -1) $
- Right: Horizontal line $ y = 2 $ for $ x > -2 $, open at $ (-2, 2) $

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4.

$$
f(x) =
\begin{cases}
-1 & \text{if } x \leq -1 \\
1 & \text{if } -1 < x < 1 \\
x & \text{if } x > 1
\end{cases}
$$

#### Step-by-step:
- For $ x \leq -1 $: $ f(x) = -1 $ → horizontal line at $ y = -1 $, closed at $ x = -1 $ → closed circle at $ (-1, -1) $
- For $ -1 < x < 1 $: $ f(x) = 1 $ → horizontal line at $ y = 1 $, but not including endpoints.
- Open circles at $ x = -1 $ and $ x = 1 $
- For $ x > 1 $: $ f(x) = x $ → line with slope 1, starting just after $ x = 1 $
- Open circle at $ (1, 1) $, then draw $ y = x $ for $ x > 1 $

#### Graph Description:
- Left: $ y = -1 $ for $ x \leq -1 $, closed at $ (-1, -1) $
- Middle: $ y = 1 $ between $ x = -1 $ and $ x = 1 $, open at both ends
- Right: $ y = x $ for $ x > 1 $, open at $ (1,1) $

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5.

$$
f(x) =
\begin{cases}
-x + 2 & \text{if } x \leq 0 \\
\frac{1}{2}x + 3 & \text{if } x > 0
\end{cases}
$$

#### Step-by-step:
- For $ x \leq 0 $: $ f(x) = -x + 2 $
- This is a line with slope $-1$, y-intercept $2$
- At $ x = 0 $: $ f(0) = -0 + 2 = 2 $ → closed circle at $ (0, 2) $
- Example points: $ (0,2), (-1,3), (-2,4) $
- For $ x > 0 $: $ f(x) = \frac{1}{2}x + 3 $
- Slope $ \frac{1}{2} $, y-intercept $3$
- But only for $ x > 0 $, so no point at $ x = 0 $
- At $ x = 0 $, $ f(0) = 3 $, but not included → open circle at $ (0, 3) $
- Then draw line increasing slowly: $ (2,4), (4,5) $, etc.

#### Graph Description:
- Left: Line $ y = -x + 2 $ from left to $ (0,2) $, closed
- Right: Line $ y = \frac{1}{2}x + 3 $ starting from $ (0,3) $, open, going upward

Note: There's a jump discontinuity at $ x = 0 $: from $ y = 2 $ to $ y = 3 $, but not connected.

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6.

$$
f(x) =
\begin{cases}
x + 2 & \text{if } x \leq 2 \\
-\frac{1}{2}x + 4 & \text{if } x > 2
\end{cases}
$$

#### Step-by-step:
- For $ x \leq 2 $: $ f(x) = x + 2 $
- Slope = 1, y-intercept = 2
- At $ x = 2 $: $ f(2) = 2 + 2 = 4 $ → closed circle at $ (2, 4) $
- For $ x > 2 $: $ f(x) = -\frac{1}{2}x + 4 $
- Slope = $-0.5$, y-intercept = 4
- At $ x = 2 $: $ f(2) = -\frac{1}{2}(2) + 4 = -1 + 4 = 3 $, but not included → open circle at $ (2, 3) $
- Then draw decreasing line: $ (4, 2), (6, 1) $, etc.

#### Graph Description:
- Left: Line $ y = x + 2 $ up to $ (2,4) $, closed
- Right: Line $ y = -\frac{1}{2}x + 4 $ starting just after $ (2,3) $, open circle at $ (2,3) $

Note: The two pieces meet at different $ y $-values at $ x = 2 $, so there's a jump.

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✔ Summary of Key Rules for Graphing Piecewise Functions:

1. Identify domains: Determine where each piece is defined.
2. Graph each piece separately within its domain.
3. Use closed circles for inclusive endpoints ($ \leq $ or $ \geq $)
4. Use open circles for exclusive endpoints ($ < $ or $ > $)
5. Check continuity at boundary points — sometimes there’s a jump or break.

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Would you like me to generate a sketch of any of these graphs? Or perhaps provide coordinate tables for plotting?