# K12MATH012: Precalculus I

Unit 3: Polynomial and Rational Functions   Unit 2 planted seeds of ideas that are extended in Unit 3 to more complex functions. A linear function is a simple, special case of a polynomial function. In a polynomial function, the unknowns will have whole number exponents. We began with linear functions because they offered a simple model with which to explore the basic concepts of function and representations of functions. Linear functions, however, can often be solved with simple arithmetic. Unit 3 extends the function ideas to the more complex functions called polynomial and rational that are more easily solved with an algebraic approach.

A rational function involves an algebraic fraction made with a polynomial numerator and a polynomial denominator. Their graphs are more complex than polynomial functions, and the solution process is more intricate than the functions in the previous chapter.

Here you will explore concepts of functions and models as the course presents symbolic and graphical ways to investigate a number of situations. The unit begins with power and polynomial functions with special attention to quadratic functions. The middle of the unit presents graphs of polynomial functions. After that you will address rational functions and their graphs. The unit closes with the inverses of polynomial functions, which are called radical functions.

This is a power-packed unit with complex concepts, so take a deep breath and be ready to think powerfully and work hard. Let’s get going!

Completing this unit should take approximately 12 hours and 45 minutes.

☐   Subunit 3.1: 2 hours and 15 minutes

☐   Subunit 3.1.1: 35 minutes

☐   Subunit 3.1.2: 1 hour and 40 minutes

☐   Subunit 3.2: 2 hours and 20 minutes

☐   Subunit 3.2.1: 30 minutes

☐   Subunit 3.2.2: 20 minutes

☐   Subunit 3.2.3: 1 hour and 30 minutes

☐   Subunit 3.3: 3 hours and 45 minutes

☐   Subunit 3.3.1: 25 minutes

☐   Subunit 3.3.2: 30 minutes

☐   Subunit 3.3.3: 40 minutes

☐   Subunit 3.3.4: 20 minutes

☐   Subunit 3.3.5: 1 hour and 50 minutes

☐   Subunit 3.4: 2 hours and 55 minutes

☐   Subunit 3.4.1: 40 minutes

☐   Subunit 3.4.2: 45 minutes

☐   Subunit 3.4.3: 1 hour and 30 minutes

☐   Subunit 3.5: 1 hour and 30 minutes

Unit3 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Identify characteristics of polynomial functions.
- Analyze intercepts and turning points of Polynomials.
- Develop graphs of quadratic functions.
- Describe the graphical behavior of a given polynomial function.
- Solve polynomial inequalities.
- Determine vertical and horizontal asymptotes of rational functions.
- Deduce domains and ranges of rational functions.
- Construct equations for rational functions given intercepts and asymptotes.
- Create inverses of polynomial and rational functions.
- Restrict domains of functions in order to create inverse functions.
- Analyze inversion possibilities of rational and polynomial functions.

3.1 Power Functions and Polynomial Functions   Polynomial functions can be used to model the outline of a mountain range or the depths of the ocean floor. They are used in determining some medical doses and for exploring inflation. Power functions and polynomial functions overlap in part. In that overlap are power functions that are simple forms of polynomial functions. Outside of the overlap are more complex polynomial functions and power functions with negative exponents. In this subunit, you will define the two types of functions and look at their basic characteristics.

3.1.1 Power Functions   You’ve already investigated some of the power functions as toolkit functions. They are simple forms of polynomial functions, so it’s a good place to start. In this subunit, you will not only define power functions but also delve into their graphs and long run behavior.

3.1.2 Polynomial Functions   Polynomial functions have whole number exponents on any unknowns. This subunit introduces a number of vocabulary terms that are useful in classifying polynomial functions. It identifies characteristics of the graphs of polynomial functions and their long run and short run behaviors. The subunit closes by finding intercepts and turning points.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 157 - 161 of the text. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 157. Answers are on page 161.

Completing this activity should take approximately 35 minutes.

• Web Media: YouTube: Mathispower4u: “End Behavior or Long Run Behavior of Functions” Link: YouTube: Mathispower4u: “End Behavior or Long Run Behavior of Functions” (YouTube)

Instructions: “End” behavior of a function is a description of what the function is like as x moves toward infinity or toward negative infinity. Click on the link above to view a video regarding the graphical perspective of “end” behavior.

Watching the video should take approximately 5 minutes.

• Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Complete the odd problems 1 - 33, beginning on page 162 of the text. The answers begin on page 530.

Completing this activity should take approximately 1 hour.

3.2 Quadratic Functions   Want to describe the trajectory of a ball thrown across a field? How about describing the speed of a falling object? Or the perfect shape for a television reception dish? All of these examples will use a quadratic function. You’ve studied them before, but we go into further depth in precalculus. This subunit explores quadratic functions, their symbolic representations, their graphs, and solution processes for quadratic equations.

3.2.1 Forms of Quadratic Functions   Quadratic functions have a square as the highest power on the unknown. This subunit focuses on quadratic functions and their graphs. You will examine characteristics of quadratic functions and the various algebraic forms of representing them. You will also explore their graphs and find vertices and intercepts. This subunit forms a firm base of concepts that later portions of the course will extend to more complex ideas.

Instructions: Click on the link to see a video explaining graphing procedures for quadratic functions.

Completing this activity should take approximately 10 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 163 - 165 of the text. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 165. Answers are on page 171.

Completing this activity should take approximately 20 minutes

3.2.2 Finding a Vertex of a Quadratic Function   A quadratic function has one vertex. It always represents the maximum or the minimum value of the function. For example, if you toss a ball into the air, the path of the ball can be modeled with a quadratic function. The vertex of that function would be at the highest point the ball traveled.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 166 - 168 of the text. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 167. Answers are on page 171.

Completing this activity should take approximately 15 minutes.

Instructions: Click on the link for an overview of finding a vertex and the roots of a quadratic function.

Watching this demonstration should take approximately 5 minutes.

3.2.3 Finding Intercepts of a Quadratic Function   A quadratic function can have up to three intercepts - one vertical intercept and two horizontal intercepts. Somewhat surprisingly, it may have only a vertical intercept and no horizontal intercepts. This subunit builds skills in finding the intercepts both algebraically and graphically.

3.3 Graphs of Polynomial Functions   Polynomial functions have complex graphs, so you will have more time to explore their attributes. This subunit addresses the intercepts of polynomial functions and the behavior found at these intercepts. It gives a bigger picture of some mathematical relationships and helps you connect ideas between the mathematics of the functions and the graphs of the functions.

3.3.1 Intercepts of Polynomial Functions   Polynomial functions have only one vertical intercept, but they can have many horizontal intercepts. This subunit teaches you how to find intercepts of polynomial functions. The horizontal intercepts are sometimes called roots or zeros.

• Checkpoint: mathcentre: “Polynomial Functions: Exercises” Link: mathcentre: “Polynomial Functions: Exercises” (Flash)

Instructions: Complete the interactive problems to check your understanding of roots of polynomials and multiplicity.

Completing this activity should take approximately 15 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 176 - 177. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 177. Answers are on page 184.

Completing this activity should take approximately 10 minutes.

3.3.2 Graphical Behavior at Intercepts   A polynomial function has only one vertical intercept, because otherwise it would not be a function. It can, however, have multiple horizontal intercepts. At a horizontal intercept, the function may change sign or loop back into the same vertical half of the coordinate grid. Mathematicians, scientists, businesspeople, and others develop functions to model real-world behavior, such as the trajectory of a meteor, the growth of income in a company, the intensity of earthquakes, and climate change. A function model makes it possible to study phenomena and to better predict future outcomes.

• Web Media: Vimeo: Jefferson College: “Graphs of Polynomial - End Behavior, Zeros, Multiplicity” Link: Vimeo: Jefferson College: “Graphs of Polynomial - End Behavior, Zeros, Multiplicity” (Flash) (HTML)

Instructions: Click on the link above for a view of how to discern characteristics of a function that determine important characteristics of its graph.

Completing this activity should take approximately 15 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 177 - 179. As you complete the reading, use pencil and paper to work through the examples.

Completing this activity should take approximately 15 minutes.

3.3.3 Solving Polynomial Inequalities   Solving a polynomial inequality means thinking deeply about the rules of signs and how the sign of each factor makes a difference in the sign of a final product. For example, if you multiply a*b*c, the product will be positive only if all factors are positive or exactly two factors are negative. The solution process that you will see in this subunit uses sign changes in multiplication in a clever way. You can also investigate the solution of a polynomial function by graphing - remember that “greater than” for a function means to look above the graph vertically, not horizontally.

• Explanation: lkinnel’s “Solving a Polynomial Inequality” Link: lkinnel’s “Solving a Polynomial Inequality” (YouTube)

Instructions: Click on the link for an overview of solving polynomial inequalities.

Completing this activity should take approximately 5minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 179 - 181. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 181. Answers are on page 181.

Completing this activity should take approximately 35 minutes.

3.3.4 Writing Equations Using Intercepts   Now we work in the other direction. Instead of starting with an equation and finding the intercepts, you will start with intercepts and find an equation that has those same characteristics. It will be helpful if you keep in mind the idea of it being a reverse process.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 181 - 182. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 182. Answers are on page 184.

Completing this activity should take approximately 20 minutes.

3.3.5 Evaluating Extrema   Quadratic functions have one maximum or minimum, but only one or the other. Polynomial functions of higher degree can have multiple maximums and/or minimums, depending upon the degree of the function. This subunit considers how to find the location of those extrema and how to evaluate a function at those points.

• Explanation: Boundless: “Relative Minimums and Maximums” Link: Boundless: “Relative Minimums and Maximums” (HTML)

Instructions: The maximum and minimum values of a function are sometimes called the global or absolute maximum and global or absolute minimum. We also address things called relative maximums and minimums. The difference among the four terms is nicely clarified in the diagram shown on this webpage. Click on the link for a helpful visual demonstration of the four terms: maximum, minimum, relative maximum, and relative minimum. While the diagram is helpful, you may want to skip the reading, as it could be confusing at this time.

Completing this activity should take about 5 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 182 - 184. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 183. Answers are on page 184.

Completing this activity should take approximately 15 minutes.

• Activity: Absolutely relative? Exploring the Concept of Maximum Instructions: A maximum is also sometimes called a global maximum or an absolute maximum. Is every absolute maximum also a relative maximum? Write a paragraph explaining why or why not, and include a diagram. Share your explanation with someone to help you edit for clarity.

This activity should take about 30 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Complete the odd problems 1 - 51, beginning on page 185 of the text. The answers begin on page 531.

Completing this activity should take approximately 1 hour.

3.4 Rational Functions   Rational Functions look like fractions, and their graphs are complex. Brushing up on your earlier fraction skills will help enormously in this subunit because algebraic fractions work in a parallel way - like when you need a common denominator. This subunit begins by defining a rational function and identifying its characteristics. It continues by exploring graphs of rational functions and helps you determine asymptotes and intercepts.

3.4.1 Rational Functions   When children first begin to divide, they often separate things into groups. For example, 6 ÷ 3 could be modeled by putting six cookies into three equal groups and counting how many cookies in one of the groups, which of course would be two. Notice also that 6 ÷ 1 would indicate making one group and of course all six cookies would be there. A difficulty arises, however, if we try to divide by zero. The problem 6 ÷ 0 would require us to take the six cookies and put them in zero piles - put them nowhere. It’s impossible! In working with rational functions, it is important to avoid dividing by zero, and the domains are restricted because of this. This subunit will define rational functions and help you develop techniques to recognize the domain of a given rational function.

Instructions: Click on the link for a video introducing graphs of rational functions. Take notes as the presenter shows how to use number sense to develop the graph of a complicated function. What part of the function gives the clue to the location of the horizontal asymptotes? How do you determine the vertical asymptotes? We can't divide by zero, but notice how the presenter uses the calculator to determine what happens as the denominator of the algebraic fraction gets closer and closer to zero. You may be wondering why the presenter doesn’t just put the function into a graphing calculator and let the machine do the thinking. This subunit is designed to help you develop number sense and an understanding of how to apply your mathematical knowledge to graphs and expand your comprehension of the behavior of a rational function.

Completing this activity should take approximately 25 minutes.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 188 - 191. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 190. Answers are on page 199.

Completing this activity should take approximately 15 minutes.

3.4.2 Finding Vertical and Horizontal Asymptotes   Think about the function f(x) = 1/x: When x gets large, the function gets very small. As x gets very large, the function gets smaller and smaller. The function is never equal to zero, but it gets closer and closer. We describe this by saying there is an asymptote. This subunit describes asymptotes and investigates how to find them.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 191 - 195. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problem on page 195. Answers are on page 200.

Completing this activity should take approximately 30 minutes.

• Explanation: Mathispower4u: “Determining Vertical and Horizontal Asymptotes of Rational Functions” Link: Mathispower4u: “Determining Vertical and Horizontal Asymptotes of Rational Functions” (YouTube)

Instructions: Click on the link above for a careful description of how to find vertical and horizontal asymptotes of rational functions. Pay attention to detail and take notes on the explanation.

Watching this video and taking notes should take approximately 15 minutes.

3.4.3 Finding Intercepts of Rational Functions   Rational functions are complex, and their graphs are complex. The good news is that if you can locate the asymptotes and find the intercepts, you can build a good approximation of the graph of that function. This subunit identifies characteristics of intercepts of rational functions and helps put together graphing concepts for these functions.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 195 - 199. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 195 - 198. Answers are on page 200.

Completing this activity should take approximately 30 minutes.

• Checkpoint: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Complete the odd problems 1 - 43, beginning on page 201 of the text. The answers begin on page 532.

Completing this activity should take approximately 1 hour.

3.5 Inverses and Radical Functions   As you find inverses of polynomial functions, you will need radical functions. For example, the inverse of the square function where x > 0 is a square root function, which is a radical function. Remember from earlier work that in finding an inverse, the domain often needs to be restricted so that the inverse is a function. In order for an inverse to exist, a function must be one-to-one as you determined in an earlier unit. Subunit 3.5 extends those ideas and applies them to more complex functions.

• Explanation: Lippman and Rasmussen’s *Precalculus: An Investigation of Functions* Link: Lippman and Rasmussen’s Precalculus: An investigation of Functions (PDF)

Instructions: Read pages 206 - 211. As you complete the reading, use pencil and paper to work through the examples and do the “Try It Now” problems on pages 209 - 211. Answers are on page 211.

Completing this activity should take approximately 30 minutes.