Okay, I see the rightmost segment now. This line starts at a closed circle at (3, 8) and ends at a closed circle at (9, 2). To find its equation, I'll calculate the slope using these two points. The change in y is 2 - 8 = -6, and the change in x is 9 - 3 = 6. So, the slope (m) is -6/6, which equals -1. Now, I will use the point-slope form with the point (3, 8) to find the equation: y - 8 = -1(x - 3). Simplifying this, I get y - 8 = -x + 3, and then y = -x + 11. The domain for this piece is all x-values from 3 to 9, including both endpoints, as indicated by the closed circles. I have now gathered all the necessary information to write the complete piecewise function.
- For $x < -4$, the function is $f(x) = -x - 6$.
- For $0 \le x < 3$, the function is $f(x) = 2x$.
- For $3 \le x \le 9$, the function is $f(x) = -x + 11$.
The function is undefined for the interval $[-4, 0)$, as there is no graphed line in that region.
$$
f(x) =
\begin{cases}
-x - 6 & \text{if } x < -4 \\
2x & \text{if } 0 \le x < 3 \\
-x + 11 & \text{if } 3 \le x \le 9
\end{cases}
$$
Parent Tip: Review the logic above to help your child master the concept of algebra 2 piecewise function worksheet.
