Piecewise, Absolute Value and Step Functions Practice ... - Free Printable

It looks like you've uploaded a graph with a piecewise linear function plotted on the coordinate plane. The graph consists of three connected line segments forming a "zigzag" shape.

Let’s analyze the graph and solve what is likely being asked — since no specific question was stated, I’ll assume you want to:

1. Determine the equation(s) of the lines that make up the graph.
2. Identify key features, such as domain, range, intercepts, and whether it's a function.
3. Possibly describe the behavior or find specific values (e.g., f(0), etc.).

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Step 1: Identify the Points

From the graph, we can identify the vertices (corner points) of the piecewise function:

- Point A: $(-4, -2)$
- Point B: $(1, 4)$
- Point C: $(2, -2)$
- Point D: $(5, 1)$

The graph connects these points in order:
- From $(-4, -2)$ to $(1, 4)$
- From $(1, 4)$ to $(2, -2)$
- From $(2, -2)$ to $(5, 1)$

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Step 2: Find Equations of Each Segment

#### Segment 1: From $(-4, -2)$ to $(1, 4)$

Slope $m = \frac{4 - (-2)}{1 - (-4)} = \frac{6}{5} = 1.2$

Use point-slope form:
$y - y_1 = m(x - x_1)$
Using point $(-4, -2)$:
$y + 2 = \frac{6}{5}(x + 4)$
$y = \frac{6}{5}x + \frac{24}{5} - 2 = \frac{6}{5}x + \frac{24}{5} - \frac{10}{5} = \frac{6}{5}x + \frac{14}{5}$

So, Equation 1:
$$
y = \frac{6}{5}x + \frac{14}{5}, \quad \text{for } -4 \leq x \leq 1
$$

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#### Segment 2: From $(1, 4)$ to $(2, -2)$

Slope $m = \frac{-2 - 4}{2 - 1} = \frac{-6}{1} = -6$

Using point $(1, 4)$:
$y - 4 = -6(x - 1)$
$y = -6x + 6 + 4 = -6x + 10$

So, Equation 2:
$$
y = -6x + 10, \quad \text{for } 1 \leq x \leq 2
$$

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#### Segment 3: From $(2, -2)$ to $(5, 1)$

Slope $m = \frac{1 - (-2)}{5 - 2} = \frac{3}{3} = 1$

Using point $(2, -2)$:
$y + 2 = 1(x - 2)$
$y = x - 2 - 2 = x - 4$

So, Equation 3:
$$
y = x - 4, \quad \text{for } 2 \leq x \leq 5
$$

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Step 3: Domain and Range

- Domain: All $x$-values covered: from $-4$ to $5$ → $[-4, 5]$
- Range: Look at all $y$-values:
- First segment goes from $y = -2$ to $y = 4$
- Second drops to $y = -2$
- Third rises to $y = 1$
- So minimum $y = -2$, maximum $y = 4$
- But does it cover everything? Let’s check:
- From $-4$ to $1$: $y$ goes from $-2$ to $4$ continuously → covers $[-2, 4]$
- Then from $1$ to $2$: $y$ drops from $4$ to $-2$ → still within $[-2, 4]$
- From $2$ to $5$: $y$ goes from $-2$ to $1$ → again within $[-2, 4]$
- So Range: $[-2, 4]$

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Step 4: Is It a Function?

Yes — each $x$ has only one $y$. The graph passes the vertical line test.

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Step 5: Intercepts

- x-intercept(s): Where $y = 0$
- Check each segment:
1. $y = \frac{6}{5}x + \frac{14}{5} = 0$ → $\frac{6}{5}x = -\frac{14}{5}$ → $x = -\frac{14}{6} = -\frac{7}{3} \approx -2.33$ → in $[-4,1]$ → valid
2. $y = -6x + 10 = 0$ → $x = \frac{10}{6} = \frac{5}{3} \approx 1.67$ → in $[1,2]$ → valid
3. $y = x - 4 = 0$ → $x = 4$ → in $[2,5]$ → valid
- So x-intercepts: $x = -\frac{7}{3}, \frac{5}{3}, 4$

- y-intercept: $x = 0$
- On first segment: $y = \frac{6}{5}(0) + \frac{14}{5} = \frac{14}{5} = 2.8$
- So y-intercept: $(0, 2.8)$

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Final Answer Summary

Piecewise Function:
$$
f(x) =
\begin{cases}
\frac{6}{5}x + \frac{14}{5}, & -4 \leq x \leq 1 \\
-6x + 10, & 1 \leq x \leq 2 \\
x - 4, & 2 \leq x \leq 5
\end{cases}
$$

- Domain: $[-4, 5]$
- Range: $[-2, 4]$
- Intercepts:
- x-intercepts: $x = -\frac{7}{3}, \frac{5}{3}, 4$
- y-intercept: $(0, \frac{14}{5})$
- Function? Yes
- Continuous? Yes (connected line segments)

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If your original task was something else (e.g., find $f(3)$, determine if increasing/decreasing, etc.), please let me know and I can adjust accordingly!

But based on the image, this is a complete analysis of the graph.