Unit 5 Rational Functions and Expressions

Lesson 1

Learning Focus

Understand the behavior of $f (x) = \frac{1}{x}$ for very large $x$ values and for $x$ values near $0$ .

Graph and describe the features of $f (x) = \frac{1}{x}$ using appropriate notation.

Lesson Summary

In this lesson, we learned about the function $f (x) = \frac{1}{x}$ , a rational function. We learned about the features of the function and its behavior near the horizontal and vertical asymptotes.

Lesson 2

Learning Focus

Transform the graph of $f (x) = \frac{1}{x}$ .

Write equations from graphs.

Predict the horizontal and vertical asymptotes of a function from the equation.

Lesson Summary

In this lesson, we learned to graph functions that are transformations of $f (x) = \frac{1}{x}$ . We learned that the transformations work just like other functions with horizontal shifts associated with the inputs to the function and the vertical effects associated with the outputs. Using these ideas, we also wrote equations to correspond with graphs and generalized each part of the equation in the form: $f (x) = k + \frac{b}{x - h}$ .

Lesson 3

Learning Focus

Define a rational function.

Explore rational functions, and find patterns that predict the asymptotes and intercepts.

Lesson Summary

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.

Lesson 4

Learning Focus

Write equivalent rational expressions.

Find the features of rational functions with numerators that are one degree greater than the denominator.

Lesson Summary

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.

Lesson 5

Learning Focus

Add, subtract, multiply, and divide rational expressions.

Lesson Summary

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.

Lesson 6

Learning Focus

Determine a process for graphing rational functions from an equation.

Lesson Summary

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.

Lesson 7

Learning Focus

Solve equations that contain rational expressions.

Lesson Summary

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.