Unit 3 Polynomial Functions

Lesson 1

Learning Focus

Model patterns of growth with tables, equations, graphs, and diagrams.

Make conjectures about function rates of change.

Lesson Summary

In this lesson, we modeled situations with quadratic and cubic functions with recursive and explicit equations. We learned that the rate of change of a cubic function is quadratic, the rate of change of a quadratic function is linear, and the rate of change of a linear function is constant.

Lesson 2

Learning Focus

Compare quadratic and cubic functions.

Graph cubic functions.

Lesson Summary

In this lesson, we examined the features of the cubic parent function, $f (x) = x^{3}$ . We identified anchor points and learned to use transformations to graph functions in the form $y = a {(x - h)}^{3} + k$ . We compared $f (x) = x^{3}$ and $g (x) = x^{2}$ to identify similarities and differences in domain, range, intercepts, and intervals of increase and decrease.

Lesson 3

Learning Focus

Add and subtract polynomials algebraically.

Add and subtract polynomials graphically.

Lesson Summary

In this lesson, we learned to add and subtract polynomials. We learned that the procedure used for adding and subtracting is analogous to adding whole numbers because polynomials have the same structure as whole numbers. Polynomials are added by adding like terms. When subtracting polynomials, we can avoid sign errors by adding the opposite of each term.

Lesson 4

Learning Focus

Multiply polynomials.

Raise binomials to powers.

Lesson Summary

In this lesson, we built on our understanding of area models from Math II to multiply polynomials. We learned to use either the box method or to distribute each term of the first factor to each term of the second factor. Both methods are based on the Distributive Property. We also learned an efficient method for raising binomials to powers using Pascal’s Triangle to help find the coefficient of each term in the expansion.

Lesson 5

Learning Focus

Divide polynomials.

Write equivalent multiplication statements after dividing.

Know when one polynomial is a factor of another polynomial.

Lesson Summary

In this lesson, we learned that polynomials can be divided using long division like whole numbers. We learned to use technology to check our work and to avoid errors in subtraction by adding the opposite of the terms to be subtracted. We found that, like numbers, a polynomial is a factor of another polynomial if it divides with no remainder. We learned two ways to write equivalent multiplication statements when there was a remainder after dividing.

Lesson 6

Learning Focus

Support or challenge claims about the result of adding, subtracting, multiplying, and dividing polynomials.

Lesson Summary

In this lesson we examined claims about the closure of sets of numbers and classes of functions under the operations of addition, subtraction, multiplication, and division. An example of such a claim is: The set of whole numbers is closed under division. A counterexample that shows this claim to be false is: $2 \div 7 = \frac{2}{7}$ . Since $\frac{2}{7}$ is a rational number, this example shows that dividing two whole numbers does not always result in a whole number.

Lesson 7

Learning Focus

Find roots and factors of quadratic and cubic functions.

Write quadratic and cubic equations in factored form.

Identify multiple roots of quadratic and cubic functions.

Lesson Summary

In this lesson, we found roots of cubic functions using the same methods we learned for quadratic functions. We found that cubic functions can have multiple roots, like quadratic functions. We learned to verify roots and write equivalent equations in factored and standard form. During the lesson, we applied the Fundamental Theorem of Algebra to cubic functions to consider the number and types of possible roots.

Lesson 8

Learning Focus

Break down a polynomial function to find the roots.

Write a polynomial function in factored form.

Lesson Summary

In this lesson, we learned to find the roots of a polynomial by using long division and factoring to break down the polynomial into factors of lower degree. Technology was useful for checking work and finding real roots that can be used to find complex roots. We determined the number and type of roots possible for each polynomial, which will be useful in writing equations and finding roots in upcoming lessons.

Lesson 9

Learning Focus

Find patterns in the end behavior of polynomial functions.

Describe the end behavior of a function using appropriate notation.

Lesson Summary

In this lesson, we examined the end behavior of polynomial and exponential functions. We found patterns that allow us to predict the end behavior for polynomials. We learned to use notation to write the end behavior for functions. We learned the definition of odd and even functions and how to identify them by their symmetries.

Lesson 10

Learning Focus

Combine pieces of information about polynomials to write equations and graph them.

Identify features of polynomials from equations and graphs.

Lesson Summary

In this lesson, we put together everything we have learned in the unit to write equations and graph polynomials. We wrote equations given roots and found roots given equations using the relationship between roots and factors. We graphed polynomials using the roots and end behavior to predict the shape of the curve.