Modeling with Rational and Piecewise-Defined Functions Module Overview
Modeling with Rational and Piecewise-Defined Functions Module Overview
This module begins with a focus on rational functions. We will explore and analyze the unique characteristics of rational functions, identifying excluded values, asymptotes, holes, domain, range and end behavior. Then, we will transition into absolute value and piecewise functions. Although there are two parts in this module, there will be ONE comprehensive assessment to evaluate your knowledge and understanding.
Part 1: Rational Functions
Many people have an interest in pastimes such as diving, photography, racing, playing music, or just getting a tan. Other people are considering a career in medicine, machinery, farming, banking, or weather. Rational functions have expressions written as a fraction. Pressure in diving, exposures in photography, average speed in racing, frequency in music and the sun's radiation can all be expressed as a radical or rational function. The careers listed are a few examples of jobs that use radical or rational functions in various aspects. Rational functions will be explored here. In addition, we will explore absolute value functions and piecewise functions. Piecewise functions are used to describe situations in which a rule or relationship changes as the input value crosses certain "boundaries."
Essential Questions
- How are rational expressions multiplied and divided?
- How are rational expressions added and subtracted?
- How are complex fractions simplified?
- How are rational equations solved?
- Why are all solutions not necessarily the solution to an equation? How can you identify these extra solutions?
- How are rational functions graphed?
- In solving rational and radical equations, what are extraneous solutions?
- What are the key features of the graphs of rational functions?
Key Terms
Rational Expression - An expression that can be written as a fraction.
Excluded Values - Values that make the expression undefined (0 in the denominator).
Like Terms - Terms having the exact same variable(s) and exponent(s).
Extraneous Solutions - Solutions that make the expression undefined.
Intercepts - Points where a graph crosses an axis.
Domain - The set of x-coordinates of the set of points on a graph; the set of x-coordinates of a given set of ordered pairs. The value that is the input in a function or relation.
Range - The y-coordinates of the set of points on a graph. Also, the y-coordinates of a given set of ordered pairs. The range is the output in a function or a relation.
Zeros - The roots of a function, also called solutions or x-intercepts.
Asymptotes - Vertical and horizontal lines where the function is undefined.
Extrema - Maximums and minimums of a graph.
End Behavior - The rise or fall of the ends of the graph.
Conjugate - The same binomial expression with the opposite sign.
Greatest Common Factor - Largest expression that will go into the terms evenly.
Lowest Common Denominator - Denominator that is the smallest multiple of all of the denominators.
Part 2: Absolute Value and Piecewise Functions
Many people have an interest in pastimes such as diving, photography, racing, playing music, or just getting a tan. Other people are considering a career in medicine, machinery, farming, banking, or weather. Rational functions have expressions written as a fraction. Pressure in diving, exposures in photography, average speed in racing, frequency in music and the sun's radiation can all be expressed as a radical or rational function. The careers listed are a few examples of jobs that use radical or rational functions in various aspects. Rational functions will be explored here. In addition, we will explore absolute value functions and piecewise functions. Piecewise functions are used to describe situations in which a rule or relationship changes as the input value crosses certain "boundaries."
Essential Questions
- How are rational expressions divided?
- How are rational functions graphed?
- What are the key features of the graphs of rational functions?
- How can prior knowledge of functions be used to build precise and efficient models?
- How can we find inputs and outputs of step functions in real world situations?
- How can we write absolute value functions as a piecewise function?
Key Terms
Absolute Value - The absolute value of a number is the distance the number is from zero on the number line.
Function - A rule of matching elements of two sets of numbers in which an input value from the first set has only one output value in the second set.
Graph of a Function - The set of all the points on a coordinate plane whose coordinates make the rule of function true.
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