Polynomial and Rational Functions


 Corey Bradley
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1 Polynomial and Rational Functions Quadratic Functions Overview of Objectives, students should be able to: 1. Recognize the characteristics of parabolas. 2. Find the intercepts a. x intercepts by solving a quadratic equation b. y intercept c. Recognize the connection between finding zeros, roots, x intercepts, and solutions to a quadratic function. 3. Find the vertex of a quadratic function Main Overarching Questions: 1. How can you find the x intercepts of a quadratic function on the graph and from the equation? 2. How can you find the y intercepts of a quadratic function from the graph and from the equation? 3. How can you find the vertex of a quadratic function from the graph and from the equation? 4. How can you determine whether the vertex of a quadratic function represents a minimum or a maximum value? 5. How can you tell whether the graph of a quadratic function should open up or down given its equation? 6. Explain how to find the zeros of a quadratic function. a. By observation in standard form b. Use the formula to find the vertex 4. Graph quadratic functions Objectives: Activities and Questions to ask students: Recognize the characteristics of parabolas. Give a few graphs of quadratic functions (that both open up and down) and the equations in 2 ax + bx + c form. Ask students to describe differences in the graphs. Why does a graph open up? Open down? When does the parabola have a maximum point? When does the parabola have a minimum point? How can we predict if a max/min exists by looking at the equation? Tell the students the max/min is always called the vertex. What kind of symmetry does the graph have? Find the intercepts Draw a parabola on the board (or provide a graphed parabola) and ask students to identify
2 i. x intercepts by solving a quadratic equation ii. y intercept iii. by looking at graph the y intercept. When the graph crosses the y axis, what is the value of x? Provide a quadratic function, and ask students to find the y intercept (plug in zero for x). Provide a few additional examples for independent or small group practice. After students have found solutions, ask them if there is a shortcut for determining the y intercept of a quadratic function (just use the c value). Using the original parabola, ask students to identify the x intercepts. When the graph crosses the x axis, what is the value of y? Provide a quadratic function, and ask students to find the x intercepts (plug in zero for y, or f(x) ). Ask students to recall methods for solving quadratic equations from previous lessons (factoring, completing the square, quadratic formula). Which method would be most efficient for this problem? Emphasize the idea that finding the x intercepts of a quadratic function is the same as solving a quadratic equation. This is also called finding the zeros, finding the roots, or finding the solutions of a quadratic function. Guide students through several additional examples, asking them to determine which method to use for each. Provide a problem set for independent practice. **Possible graphing calculator usage: After students have identified intercepts using the equations, allow them to graph the functions and check their answers using the calculate: value and calculate: zero features. Find the vertex of a quadratic function Introduce standard form: a ( x h) + k. Then, give several graphs of quadratic functions o By observation in standard form with their standard form equations and vertices plotted. o Use the formula to find the vertex Have students examine the differences in each situation and determine a pattern. Can the equation give us the vertex? How? 2 Now give the equation in ax + bx + c form. How would we find the vertex this time? Give the formula to find the x coordinate of the vertex. Have students practice finding the x coordinate. How would we find the y coordinate? Graph quadratic functions Have student students list the properties they would need to graph a quadratic function (vertex, x intercepts, y intercept, and domain and range). Have students describe in their own words how to find these properties. Give students a worksheet with a few quadratic equations to graph.
3 Polynomial Functions Overview of Objectives, students should be able to: 1. Identify polynomial functions Main Overarching Questions: 1. How do you identify properties of a polynomial function s graph and use them to graph? 2. Recognize characteristics of graphs of polynomial functions. 3. Understand the relationship between degree and turning points 4. Determine end behavior 5. Use factoring to find zeros of polynomial functions 6. Identify zeros and their multiplicities 7. Graph polynomial functions Objectives: Activities and Questions to ask students: Identify polynomial functions Ask students to define a polynomial. What do you think of when you hear the word? Some may say powers multiple terms or at least three terms Give students a few examples of polynomials and a few examples of non polynomials (rational functions, radical functions, and exponential functions). What is the difference between the polynomial and non polynomial functions? Recognize characteristics of graphs of polynomial functions To give students an idea of what polynomial graphs will look like, give a few polynomial functions graphs. Then, compare with non polynomial function graphs that have breaks and sharp turns. Have students compare. What are some traits of a polynomial function s
4 Understand the relationship between degree and turning points graph? (Smooth turns, peaks and valleys, and is infinite in both directions) Return to the several graphs of polynomials presented before. Include the function equations with each graph. Ask students to count the number of turning points in each graph. If further discussion of what a turning point is necessary explore the concept. If your examples are chosen wisely so that all have n 1 turning points, some students may see the connection between degree and the number of turning points. If not, ask students to determine the degree of each polynomial function and THEN determine the relationship. You will need to point out that functions can have at MOST n 1 turning points, but can certainly have less. Determine end behavior Have a discussion about what end behavior means to the students. Give an example of a polynomial function graph. Does the function end? Does it go up or down? On what side does it go up or down? Explain that end behavior refers to what the graph does far to the right (for large x) and far to the left (for small x). Refer to the earlier graphs. What is the end behavior? What two choices are there? Students should see that polynomial functions either increase without bound or decrease without bound on either side of the graph. Begin with two simple polynomials and their graphs. For example: f (x) = x 4 + x 2 and g(x) = x 4 + x 2. Ask students to compare the end behavior in each case. Can the equation determine the end behavior? Why did one graph go up on the left and up on the right but the other graph goes down on the left and down on the right? What change do you see in the equation? Have students draw the conclusion that the sign of the leading coefficient determines part of the end behavior. Students may think this is all they need to know. Now, give an example of an odd and even degree polynomial (keep the leading coefficient positive in both cases). f (x) = x 4 + x 2 and g(x) = x 3 + x 2. Again ask students to compare the end behavior of each graph? Why does one graph go up on the left and up on the right but the other graph goes down on the left and up on the right.
5 Use factoring to find zeros of polynomial functions Have students draw the conclusion that the degree of the polynomial (odd or even) determines part of the end behavior. Summarize the 4 cases and have students practice determining end behavior. This is an extension of finding zeros of quadratic functions. Simply ask students how they found zeros of quadratic functions. Hopefully most will remember to set the function to zero and solve. If needed discuss what a zero is on the graph of a function. Why do we need it? What can the zeros tell us? How do we solve polynomial equations? What was the easiest way to solve quadratic equations? Factoring! Have students practice finding zeros. Identify zeros and their multiplicities This concept can be difficult for students to see on their own. Consider giving a factored polynomial function with zeros and their multiplicities given. You might need to give several examples. Consider using multiplicities 2 and greater and avoid a zero of 0. Ask students what they think multiplicity means. You can mention that the equation tells us the multiplicity of each zero. Hopefully students will see the relationship between the power on the factor and the multiplicity. If not, you can clue them in. Next, give the graph of a polynomial function with some zeros that cross the x axis and some that just touch the axis. Ask students to compare the zeros. How do they behave differently? Give a few graphs like this if necessary. Next, beside each zero include its multiplicity. Give several examples and let students consider if there is a relationship between the multiplicity and the behavior of the zero. Have students summarize the relationship between zeros multiplicities and the behavior of the zero. Graph polynomial functions This is an important point to stop and summarize all the properties studied thus far. Have students list all the properties studied and how we can use the equation of the function to describe these properties. With student involvement practice graphing a polynomial function. It s important to stress the work needed to describe the properties before even graphing the function.
6 Dividing Polynomials Overview of Objectives, students should be able to: 1. Use long division to divide polynomials Main Overarching Questions: 1. How do you divide polynomials? 2. Use synthetic division to divide polynomials Objectives: Activities and Questions to ask students: Use long division to divide polynomials Ask students to next consider what happens if we have more than one term in the 2 x 7x+ 12 denominator like. Can we split up the denominator? If students say yes, ask x 5 them to combine together 2 3 x + 5 Ask them once again to consider if they can split up the denominator. Once students are convinced that they cannot split the denominator, suggest using long division to divide. Give students all the terminology before beginning. As a good analogy, complete the steps to long division concurrently with an example from arithmetic. Have students compare/contrast each step of polynomial long division with the arithmetic version. Give students a few long division problems to try on their own. Use synthetic division to divide polynomials Before explaining all the steps, consider showing a long division problem fully worked out and by its side the same division completed using synthetic division. Have students compare the two solutions. Where did the number in the box come from? Where did the first row of numbers come from? The 2 nd row? The final row under the line? Where is the remainder? After students realize where the coefficients come from, illustrate the process of working
7 through the synthetic division. Have students summarize the process after completing an example. Are there any drawbacks to this method? What if the divisor isn t of the form x c? Have students practice this new method. Rational Functions Overview of Objectives, students should be able to: 1. Find the domains of rational functions Main Overarching Questions: 1. How do you identify properties of a rational function s graph and use them to graph? 2. Use arrow notation 3. Identify vertical asymptotes 4. Identify horizontal asymptotes 5. Use transformations to graph rational functions 6. Graph rational functions 7. Identify slant (oblique) asymptotes Objectives: Activities and Questions to ask students: Find the domains of rational functions Already covered in earlier lesson, but a review might be warranted. Use arrow notation By plotting points, have students graph f (x) = 1 x. What happens as x gets close to zero? Some students may say goes up while others may say go down. Discuss why both answers are correct. Then, ask what happens as x approaches zero on the left side. Once
8 Identify vertical asymptotes students answer, give the arrow notation that shows from the left As x 0, f (x). Repeat on the right side. Give them some examples to try. For the function f (x) = 1 x, what happens at x = 0? Does the function exist? What does the graph do at x = 0. Explain that at x = 0, the graph has a vertical asymptote defined as the vertical line x = 0. Give a few other graphs of rational functions along with their function equations that are of the form f (x) = 1 x c. Is there a relationship between the vertical asymptote and the equation of the function? What is the value of the denominator at the vertical asymptote value? Have students describe how they would find the vertical asymptote when given a function. If students insist on observing the value, give a more complicated denominator that requires solving or factoring then solving. How would we find the vertical asymptote for this function? Identify horizontal asymptotes Give three examples of the graphs and function equations that do not have a horizontal asymptote, three examples of a non zero horizontal asymptote rational function, and three examples of a zero horizontal asymptote rational function. Begin by asking students to determine the end behavior in each case. End behavior was discussed with polynomial functions, so students should have an understanding of what end behavior means. Is there a relationship between the differences in the degree of the numerator and denominator and the end behavior? What happens if the degree of the numerator is greater than the degree of the denominator? What happens if the degree of the numerator is equal to the degree of the denominator? What happens if the degree of the numerator is less than the degree of the denominator? Have students establish the relationship. Give a few examples to try. Use transformations to graph rational functions Transformation was covered earlier. Have students practice transforming the functions f (x) = 1 x and g(x) = 1 x 2. Graph rational functions Before graphing, ask students how they would find the zeros of a rational function. If we set the function equal to zero, what part of the fraction must be zero? Have students realize the numerator must be set to zero to find the zeros of a rational function. Now, have students list the properties and how to get them of rational functions. Stress the
9 importance of finding all this information and using it to find the graph. Have students practice finding all the properties and using them to graph. Identify slant (oblique) asymptotes Ask students about the end behavior when the degree of the numerator is greater than the degree of the denominator. Most will say goes up or goes down or that there is no horizontal asymptote. Explain that there are sometimes other types of end behavior asymptotes. To show this, give a graph of a rational function with a slant asymptote. Label the equation of the function and the equation of the slant asymptote. Ask students to use long division (or synthetic) to divide the numerator by the denominator. Is there a relationship between the quotient and the slant asymptote? Provide another example if necessary. To answer the question as to why the slant asymptote is the quotient (and not the remainder), ask students to give the remainder expression (not just the remainder, but the remainder divided by the divisor). What happens as x to the remainder expression? Have students try to plugin really large values of x. The remainder gets very close to zero, so the fraction is very closely approximated by the quotient, and the remainder has little to no effect. Polynomial and Rational Inequalities Overview of Objectives, students should be able to: 1. Solve polynomial inequalities 2. Solve rational inequalities Main Overarching Questions: 1. How do you find the solution to a polynomial inequality? 2. How do you find the solutions to a rational inequality? 3. Solve problems modeled by polynomial or rational inequalities. Objectives: Activities and Questions to ask students: Solve polynomial inequalities Give a simple example of a polynomial inequality like x 2 7x +10 < 0. How would we solve this? Students might have no idea, or might suggest isolating x. Ask how they would solve x 2 7x +10 = 0? Hopefully, they will suggest to factor and set each factor equal to 0.
10 Have students complete this and see that the solution is x = 2 and x = 5. Next, graph the function x 2 7x +10. For what values of x is the function below the x axis? If we are trying to solve x 2 7x +10 < 0, how can we use the graph to tell us the solution? The idea is to get students to observe that the values of x for points below the x axis are the solutions to this inequality. Repeat the discussion for x 2 7x +10 > 0. Next, discuss where the boundary points where the function moves from positive to negative, and negative to positive. Where did this happen? The zeros of the function. Give students a few more examples and have them verify that the only place where the functions change sign are at the zeros. Introduce the idea of a sign chart to the students and show them how we can use the chart to solve the inequality. Have students summarize the process of solving polynomial inequalities. Solve rational inequalities The above discussion can be repeated for rational inequalities. The students can discover that now the sign changes can occur at both zeros AND vertical asymptotes, so both types of points must be plotted on the number line. It will be important to have students work some examples with, to understand when to include the points (zeros) and to not include the points (the vertical asymptotes). Solve problems modeled by polynomial or rational inequalities. An interesting application from Blitzer s College Algebra introduces the quadratic equation: s(t) = 16t 2 + v 0 t + s 0. One exercise has students set up and solve the quadratic inequality in this situation: Divers in Acapulco, Mexico, dive headfirst at 8 feet per second from the top of a cliff 87 feet about the Pacific Ocean. During which time period will the diver s height exceed that of the cliff? Again, the key here is to be able to setup the problem. What values go in for what variables? Which variable are you solving for? The contents of this website were developed under Congressionally directed grants (P116Z090305) from the U.S. Department of Education. However, those contents do not necessarily represent the
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