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.

• Closed Circle: Indicates the endpoint is included in the interval (use ≤ or ≥).

• Open Circle: Indicates the endpoint is not included in the interval (use < or >).

• Equation of a Line: , where is the slope and is the y-intercept.

Step-by-Step Guidance

1. Identify the intervals for each segment. Look at the x-values where each line segment starts and ends, and note whether the endpoints are open or closed circles.

2. For each segment, find the equation of the line. Use the two endpoints of each segment to calculate the slope and then use point-slope form to find the equation.

3. Write the piecewise function using the equations you found, making sure to use the correct interval notation (open or closed) based on the circles in the graph.

4. Double-check that your piecewise function matches the graph for all intervals and endpoints.

Try solving on your own before revealing the answer!