Step-by-Step Guidance for Calculus Midterm Study Prep (Part 1)

Q2. Find the piecewise formula of the graph below.

Background

Topic: Piecewise Functions

This question tests your ability to interpret a graph and write the corresponding piecewise-defined function. Piecewise functions are defined by different expressions depending on the interval of the input variable (x).

Key Terms and Formulas:

• Piecewise Function: A function defined by multiple sub-functions, each applying to a certain interval of the domain.

• Interval Notation: Used to specify the range of x-values for each piece.

• Open and Closed Circles: On a graph, a closed (filled) circle means the endpoint is included (≤ or ≥), while an open circle means the endpoint is not included (< or >).

Step-by-Step Guidance

1. Examine the graph and identify the intervals for which each line segment is defined. Pay attention to open and closed circles to determine whether endpoints are included.

2. For the first segment (orange), note the endpoints and their coordinates. Write the equation of the line passing through these points and specify the interval for x.

3. For the second segment (blue), repeat the process: identify the endpoints, determine if they are included or excluded, and write the equation of the line for this interval.

4. Combine the two equations into a piecewise function, using proper notation and interval restrictions for each piece.

Try solving on your own before revealing the answer!

Final Answer:

The piecewise function is:

$y = \begin{cases} -x + 3, & -1 \leq x \leq 2 \\ -2x + 3, & 2 < x \leq 3 \end{cases}$

The first piece corresponds to the segment from $x = -1$ to $x = 2$ (including both endpoints), and the second piece corresponds to the segment from $x = 2$ (not included) to $x = 3$ (included).