OUTLINE FOR A SAMPLE LESSON PLAN



MAT 117 WEEK 8 LESSON PLAN

Total estimated time: 140 minutes

Note: It is very important that we help students make the connections by spending a few minutes each day reviewing the most important concepts from the previous day’s lecture, especially those concepts most related to the new lecture.

Objectives

Polynomial Functions

Use the Fundamental Theorem of Algebra and the Linear Factorization Theorem to write a polynomial as the product of linear factors

Find all real and complex zeros of a polynomial function

Find a polynomial with integer coefficients whose zeros are given

• Use the Leading Coefficient Test and the zeros of a polynomial to sketch the graph of a polynomial

• Apply techniques for approximating real zeros to solve an application problem

Rational Functions

• Find the domain of a rational function

• Find the vertical and horizontal asymptotes of the graph of a rational function

• Sketch the graph of a rational function

• Use a rational function model to solve an application problem

Polynomial Functions

Motivation: [5 minutes] Solving equations is at the heart of any math course. For example, in calculus, to find the critical values students will need to solve equations. Polynomial equations are one example of such equations. Historically, students learned how to find the roots from the given polynomial. Now this can often be done much more easily with the graphing calculator. But the reverse process is also interesting; that is, from the given graph of a polynomial, recover a formula for it.

In addition, polynomials are used frequently to model real-life scenarios and make predictions. Polynomial models are often simpler than other models. Solving polynomial equations often leads to simple solutions for real-life applications.

Warm Up Discussion: [5 minute] Provide an example of a polynomial function (in completely factored form) with only real roots. Ask students how many roots it has. Then provide an example of another polynomial function (in completely factored form) with mixed real and complex roots. Ask students how many real roots and how many complex roots it has. Then help students generalize that an nth degree polynomial has n roots.

1. Find all real and complex zeros of a polynomial function

Warm Up Example or Activity: [20 minutes] Give a polynomial of degree 3 with integer coefficients and one rational root. Use the rational root theorem and synthetic division to find that root. Then use the quadratic formula to find the other complex roots. Point out that complex roots occur in complex conjugate pairs. Point out that this polynomial can be completely factored as a product of three factors. (Then use the graphing calculator to demonstrate that the graph has just one real root; the complex roots do not show as x intercepts.)

Formal Concept: [5 minutes] The Fundamental Theorem of Algebra and the Linear Factorization Theorem can be used to write a polynomial as the product of linear factors.

2. Find a polynomial with integer coefficients whose zeros are given

Warm Up Example or Activity: [20 minutes] Ask students to find a formula for a degree four polynomial with integer coefficients that has two real zeroes and one complex zero (a + bi, with b ( 0). Demonstrate that this polynomial also has the other complex conjugate as a root. Explore different possible solutions, based on the leading coefficient. Have the students graph the functions and observe how changing the sign of the leading coefficient from positive to negative changes the global behavior. (In the previous example, with n = 3, an odd degree, explore how changing the sign of the leading coefficient would change the global behavior of the 3rd degree polynomial.)

Formal Concept: [5 minutes] Have students generalize the Leading Coefficient Test in their own words.

3. Apply techniques for approximating real zeros to solve an application problem

Example and In-Class Activity: [10 minutes] Have students solve a problem involving a 2nd degree or higher polynomial model for revenue, cost, or profit.

Suggested Follow Up Assessment: the assigned homework problems, and a quiz or warm-up review example at the beginning of the next class.

Rational Functions

Motivation: [5 minutes] Some things grow with limited capacity because of limited space or resources, such as a fish population in a pond. Other things cannot realistically reach 100% optimization, such as pollution removal. Other things decrease over time, such as the concentration of medicine or alcohol in the bloodstream. Rational functions can be used to model these situations and also are used with limits and applications in calculus.

Warm Up Discussion: [5 minutes] One of the most important aspects of rational functions is the concept of vertical and horizontal asymptotes. The graphs of rational functions often are in pieces, with vertical asymptotes (local behavior) at places where the input is not defined and horizontal asymptotes (global behavior). Horizontal asymptotes demonstrate the limiting capacity in applications of rational functions.

1. Find the domain of a rational function.

Find the vertical and horizontal asymptotes of the graph of a rational function

Warm Up Example or Activity: [20 minutes] Choose a rational function of degree one over degree one in completely simplified form. Ask students for the domain. Remind them that the domain of the function is those real values of x that make the function have meaning. Pick some x values in the domain and the number that is not in the domain. Then talk about the presence of a vertical asymptote on the graph at that x value. Ask students to graph the function to verify this. Demonstrate for them the behavior to the left and right of this value.

Also, ask the students to zoom out to demonstrate the global behavior of the function. Discuss the equation of the horizontal asymptote. Give other quick examples of other cases for horizontal asymptotes, i.e., when the horizontal asymptote is zero or when there is no horizontal asymptote.

Formal Concept: [5 minutes] Have students explain in their own words how to find the domain and the vertical asymptote of a rational function algebraically. Also, lead them to examine and state in their own words how the ratio of the leading terms of the polynomials in the numerator and denominator is related to the equation for the horizontal asymptote.

2. Sketch the graph of a rational function

Warm Up Examples or Activities: [20 minutes] Give students more examples of higher degree polynomials in the numerator and the denominator to help the students learn to

a) find the y and x intercepts and the domain

b) find the equations of the vertical and horizontal asymptotes

c) select some extra x values to aid in graphing (choose values between vertical asymptotes and the x intercept)

d) graph the function by hand and confirm using your calculator

You may choose an example where the graph intersects the horizontal asymptote locally. (Many students think that the graph cannot intersect the horizontal asymptote.) You may also choose to give an example of a denominator with no real roots and examine the effect this has on the graph.

3. Use a rational function model to solve an application problem

Warm Up Examples or Activities: [15 minutes] Choose any real applications from the book, e.g., population of animals, pollution removal, drug concentration, average cost, etc. Help students discover that the horizontal asymptote of the function is the limiting capacity (maximum population) or minimum concentration for these kinds of problems.

Suggested Follow Up Assessment: the assigned homework problems, and a quiz or warm-up review example at the beginning of the next class.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download