The function is a graph that has 2 line segments the first one has segments that stretches from (0,2) to (2,0). The last line segment starts at (2,1) and ends at (5,0). Write a piecewise formula for the function 

To express the function as a piecewise function, we need to find the equations of the two line segments described in the problem.

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

The slope \( m \) of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

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

For the first segment:

- \( (x_1, y_1) = (0, 2) \)
- \( (x_2, y_2) = (2, 0) \)

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

The equation of the line is in the form:

\[
y = mx + b
\]

Using one of the points, say \((0, 2)\), to find the y-intercept \( b \):

\[
2 = -1(0) + b \implies b = 2
\]

Thus, the equation for the first segment is:

\[
y = -x + 2 \quad \text{for} \quad 0 \leq x \leq 2
\]

### 2. **Second Line Segment: From (2,1) to (5,0)**

For the second segment:

- \( (x_1, y_1) = (2, 1) \)
- \( (x_2, y_2) = (5, 0) \)

\[
m = \frac{0 - 1}{5 - 2} = \frac{-1}{3} = -\frac{1}{3}
\]

Using the point \((2, 1)\) to find the y-intercept \( b \):

\[
1 = -\frac{1}{3}(2) + b \implies 1 = -\frac{2}{3} + b \implies b = 1 + \frac{2}{3} = \frac{5}{3}
\]

Thus, the equation for the second segment is:

\[
y = -\frac{1}{3}x + \frac{5}{3} \quad \text{for} \quad 2 \leq x \leq 5
\]

### Piecewise Function

Combining both segments, the piecewise function is:

\[
f(x) =
\begin{cases} 
-x + 2 & \text{for } 0 \leq x \leq 2 \\
-\frac{1}{3}x + \frac{5}{3} & \text{for } 2 \leq x \leq 5 
\end{cases}
\]

Would you like to go over the steps in more detail, or do you have any questions?

---

1. What is the general formula for the slope of a line?
2. How do you find the equation of a line using a point and a slope?
3. Can the equations of these line segments be represented differently?
4. What does the y-intercept represent in the context of a line equation?
5. How do we determine the domain for each segment in a piecewise function?
6. What happens to the function outside the domain of the defined segments?
7. How do you determine the continuity of the piecewise function?
8. How would the piecewise function change if the segments overlapped?

**Tip:** When dealing with piecewise functions, ensure each segment is clearly defined within its specific domain to avoid ambiguity.

Explore how to create a piecewise function from two line segments defined by points (0,2) to (2,0) and (2,1) to (5,0). Learn the slope formula, point-slope form of a line, and how to define domains in piecewise functions. Suitable for high school students.

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