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.