Graph Piecewise Functions

Algebra-1

1. Fundamental Concepts

Definition: Piecewise functions are defined by different expressions over different intervals of the domain.
Graphing: Graph each piece of the function in its respective interval and ensure the graph is continuous where the pieces meet, if applicable.
Domain: The domain of a piecewise function is the union of the domains of its individual pieces.

2. Key Concepts

Basic Rule: $f(x) = \begin{cases} 2x + 3 & \text{if } x \leq 0 \\ x^2 - 1 & \text{if } x > 0 \end{cases}$

Degree Preservation: The highest degree in the result matches input

Application: Used to model situations with different behaviors at different intervals

3. Examples

Example 1 (Basic)

Problem: Graph the piecewise function $f(x) = \begin{cases} x + 1 & \text{if } x < 0 \\ x^2 & \text{if } x \geq 0 \end{cases}$

Step-by-Step Solution:

For \(x < 0\), graph \(y = x + 1\). This is a straight line with a slope of 1 and y-intercept at (0, 1).
For \(x \geq 0\), graph \(y = x^2\). This is a parabola opening upwards with its vertex at (0, 0).
Ensure the graph is continuous at \(x = 0\).

Validation: Check points on both sides of \(x = 0\) to ensure continuity.

Example 2 (Intermediate)

Problem: Graph the piecewise function $f(x) = \begin{cases} -x + 2 & \text{if } x \leq 1 \\ 2x - 3 & \text{if } x > 1 \end{cases}$

Step-by-Step Solution:

For \(x \leq 1\), graph \(y = -x + 2\). This is a straight line with a slope of -1 and y-intercept at (0, 2).
For \(x > 1\), graph \(y = 2x - 3\). This is a straight line with a slope of 2 and y-intercept at (0, -3).
Ensure the graph is continuous at \(x = 1\).

Validation: Check points on both sides of \(x = 1\) to ensure continuity.

4. Problem-Solving Techniques

Visual Strategy: Use different colors or line styles for different pieces of the function.
Error-Proofing: Always check the boundaries of the intervals to ensure continuity.
Concept Reinforcement: Practice with various types of functions (linear, quadratic, etc.) to understand their behavior in different intervals.