Graphs with Linear Parts Practice (Piecewise) - MathBitsNotebook(A1) - Free Printable

It looks like you've uploaded a graph showing a piecewise function made up of two line segments. Let's analyze and solve the problem based on what we can see in the image.

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Step 1: Understand the Graph

The graph shows two connected line segments:

1. First segment: From point $(-3, -3)$ to $(1, 2)$
2. Second segment: From $(1, -1)$ to $(3, -3)$

Note:
- There is an open circle at $(1, -1)$, meaning this point is not included in the second segment.
- The first segment ends at $(1, 2)$, which is a closed circle, so it is included.
- So, the function has a jump discontinuity at $x = 1$.

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Step 2: Determine the Function Definition

We'll define this as a piecewise function.

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

Let’s find the equation of the line through these points.

- Slope $m = \frac{2 - (-3)}{1 - (-3)} = \frac{5}{4}$
- Using point-slope form with point $(-3, -3)$:

$$
y + 3 = \frac{5}{4}(x + 3)
$$
$$
y = \frac{5}{4}x + \frac{15}{4} - 3 = \frac{5}{4}x + \frac{15}{4} - \frac{12}{4} = \frac{5}{4}x + \frac{3}{4}
$$

So, for $x \in [-3, 1]$, the function is:
$$
f(x) = \frac{5}{4}x + \frac{3}{4}
$$

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

But note: there is an open circle at $(1, -1)$, so this segment starts after $x = 1$. So the domain is $x \in (1, 3]$.

Find the slope:
$$
m = \frac{-3 - (-1)}{3 - 1} = \frac{-2}{2} = -1
$$

Use point-slope with $(3, -3)$:
$$
y + 3 = -1(x - 3) \Rightarrow y = -x + 3 - 3 = -x
$$

Wait: $y = -x$

Check: At $x = 3$, $y = -3$ → correct
At $x = 1$: $y = -1$, but that point is not included, so we define this piece for $x > 1$

So:
$$
f(x) = -x, \quad \text{for } x \in (1, 3]
$$

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Final Piecewise Function:

$$
f(x) =
\begin{cases}
\frac{5}{4}x + \frac{3}{4}, & -3 \leq x \leq 1 \\
-x, & 1 < x \leq 3
\end{cases}
$$

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Answer Summary

The graph represents a piecewise linear function defined as:

$$
\boxed{
f(x) =
\begin{cases}
\frac{5}{4}x + \frac{3}{4}, & \text{if } -3 \leq x \leq 1 \\
-x, & \text{if } 1 < x \leq 3
\end{cases}
}
$$

This function has a jump discontinuity at $x = 1$ because the left-hand limit is $f(1) = 2$, but the right-hand side starts at $x > 1$, approaching $-1$ from the right (but never reaching it).

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If your original task was to find the function rule, evaluate at certain points, or determine continuity, please let me know — I can tailor the answer further!