Unit 5 Rational Functions and Expressions
Understand the behavior of
Graph and describe the features of
In this lesson, we learned about the function
Transform the graph of
Write equations from graphs.
Predict the horizontal and vertical asymptotes of a function from the equation.
In this lesson, we learned to graph functions that are transformations of
Define a rational function.
Explore rational functions, and find patterns that predict the asymptotes and intercepts.
In this lesson, we learned to identify the horizontal and vertical asymptotes of a rational function by comparing the degree of the numerator to the degree of the denominator. The vertical asymptotes occur where the function is undefined, and the horizontal asymptote describes the end behavior of the function. Finding the intercepts is the same as other functions we know but there are ways to be more efficient with rational functions.
Write equivalent rational expressions.
Find the features of rational functions with numerators that are one degree greater than the denominator.
In this lesson, we learned that equivalent expressions can be found for rational expressions like rational numbers when there are common factors in the numerator and denominator. When the degree of the numerator is greater than the degree of the denominator, we learned that a rational expression can be written in an equivalent form by dividing the numerator by the denominator. When this operation is performed on a rational function, the quotient indicates the end behavior or slant asymptote of the function.
Add, subtract, multiply, and divide rational expressions.
In this lesson, we learned that performing operations on rational expressions is just like performing operations on rational numbers. Multiplication is performed by multiplying the numerators together, multiplying the denominators together, and dividing out any common factors. Division is performed by inverting the divisor and then multiplying the two fractions. Addition and subtraction require obtaining a common denominator and then combining the numerators into one fraction with the common denominator.
Determine a process for graphing rational functions from an equation.
In this lesson, we learned to sketch graphs of rational functions by finding the intercepts and the asymptotes, and by determining the behavior near the asymptotes. From this information we sketched the general shape of the graph without calculating exact points.
Solve equations that contain rational expressions.
In this lesson, we learned several strategies for solving rational equations. We found that it is often useful to combine two fractions into one expression or to multiply both sides of the equation by the common denominator of the fractions. Solving rational equations sometimes produces an extraneous solution that makes the denominator of one of the rational expressions in the original equation equal to zero and is therefore not an actual solution to the equation.