Pre-Calculus Prep: Graphing Piecewise Functions on a Coordinate Plane

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Pre-Calculus Prep: Graphing Piecewise Functions on a Coordinate Plane is a topic in pre-calculus mathematics that focuses on plotting and understanding functions that are defined by different rules or formulas over specific intervals. Piecewise functions consist of multiple "pieces" or sections, each defined by a different equation.

Here are the key concepts and ideas related to graphing piecewise functions on a coordinate plane in pre-calculus:

1. Definition of Piecewise Functions: A piecewise function is a function that is defined differently for different intervals or subdomains. It consists of multiple cases or "pieces," and each piece has its own equation or rule.

2. Steps to Graph Piecewise Functions: To graph a piecewise function, follow these steps:
a. Identify the different intervals or subdomains on which the function is defined.
b. Write down the equation or rule for each interval.
c. Plot the points for each interval by evaluating the function at specific values within that interval.
d. Connect the plotted points with line segments or curves to form the graph of the piecewise function.

3. Domain and Range: The domain of a piecewise function is the union of the domains of its individual pieces. The range is the set of all possible y-values that the function takes on within its defined intervals.

4. Discontinuities: Piecewise functions may have discontinuities at the points where the different pieces meet. These points should be indicated on the graph, and their nature (removable, jump, or infinite) should be considered.

5. Interpreting the Graph: Analyzing the graph of a piecewise function allows us to understand the behavior and characteristics of the function in different regions. We can observe how the function changes, whether it is continuous or discontinuous, and whether it exhibits particular patterns or trends within its defined intervals.

6. Real-World Applications: Piecewise functions are often used to model situations where different rules or formulas apply in different scenarios. They can represent situations such as piecewise-defined linear functions, step functions, or functions with different behaviors based on specific conditions.

Understanding how to graph piecewise functions is essential for visualizing and analyzing complex relationships and scenarios. By plotting the different pieces of the function and considering their domains, ranges, and discontinuities, we can gain insights into how the function behaves and make predictions about its values in various mathematical and real-world contexts.

These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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