The graph is a line segment that connects points (0,0) and (1,1) and another line segment that connects the other line segment at points (1,1) to (2,0). Write a piecewise formula for the function

To write a piecewise formula for the function described by the two line segments, let's determine the equation of each segment.

### 1. **First Line Segment: From (0,0) to (1,1)**

The slope \( m \) is calculated as:

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - 0}{1 - 0} = 1
\]

The equation of the line in slope-intercept form \( y = mx + b \) can be written as:

\[
y = 1x + 0 \quad \text{or simply} \quad y = x
\]

So, the equation for the first segment is:

\[
y = x \quad \text{for} \quad 0 \leq x \leq 1
\]

### 2. **Second Line Segment: From (1,1) to (2,0)**

For the second segment:

\[
m = \frac{0 - 1}{2 - 1} = \frac{-1}{1} = -1
\]

Using the point \((1, 1)\) to find the y-intercept \( b \):

\[
1 = -1(1) + b \implies b = 2
\]

Thus, the equation for the second segment is:

\[
y = -x + 2 \quad \text{for} \quad 1 \leq x \leq 2
\]

### Piecewise Function

The piecewise function that describes the graph is:

\[
f(x) =
\begin{cases} 
x & \text{for } 0 \leq x \leq 1 \\
-x + 2 & \text{for } 1 \leq x \leq 2 
\end{cases}
\]

Would you like to explore any part of this solution further, or do you have additional questions?

---

1. How do we determine the slope between two points on a line?
2. What are the different forms of linear equations?
3. Why is it important to define the domain for each piece of a piecewise function?
4. How does the continuity of the function affect the piecewise representation?
5. How would the function change if the second segment started at a different point?
6. What does it mean for a function to be continuous at a point like \( x = 1 \)?
7. How can we graph this piecewise function to verify its accuracy?
8. What happens if the segments overlap or there is a gap between them?

**Tip:** When writing piecewise functions, always ensure the equations match the behavior of the graph within their respective intervals.

Learn how to write a piecewise formula for a function described by line segments from (0,0) to (1,1) and (1,1) to (2,0). This problem covers key concepts in linear equations and piecewise functions, including the slope formula and point-slope form. Suitable for students in Grades 9-12.

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