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# Chapter 4.1 - Exercises

# Khan Academy - Vertical Asymptotes, Horizontal Asymptotes and Holes

# Introduction to Rational Functions and Asymptotes

# Section 1: Introduction to Rational Functions

A graphing calculator would be helpful. You can also use DESMOS. However, students should be able to logically determine the behavior of a rational function around the singular points and its end-behavior.

A rational function R is the ratio P/Q of two polynomials P and Q. Thus the domain of R is the set of all values of x for which Q(x) is not zero. The points ** a** for which Q(x) = 0 are called singular points. How does R behave around a singular point

**? If the values of R become unboundedly large then R has a**

*a***vertical asymptote**

**. On the other hand if R approaches a finite number then the graph of R has a**

*x - a***hole**at that singular point.

The end-behavior of R is the study of values of R when x becomes unboundedly large in either direction. If the degree of P in the numerator is less than or equal to the degree of Q in the denominator, then the values approach a finite number L. The **line y = L** is a **horizontal asymptote** to R.

If the degree of P is more than the degree of Q, then we can write R = q + r/Q, where q is the quotient on (long) dividing Q into P and r is remainder. In this case, values of R become unboundedly large along the quotient function q. If q is linear then it is called **slant asymptote**. If q is nonlinear then it is called **curve asymptote**.

**Review: **Polynomial factoring and zeros, sign chart, long division method.

**Learning Objectives:**

- Find the domain of a rational function.
- Study the behavior of the function around singular points.
- Identify the vertical asymptotes and holes.
- Study the end-behavior and identify horizontal asymptote(s) or curved/slant asymptotes any.
- Draw the graph using the above information.