1 - Power and Polynomial Functions

In Algebra 1, students worked to make connections between representations of functions. They observed that different forms of symbolic representations of linear and quadratic functions revealed different features of the functions and vice versa. Factored form, for example, revealed rational zeroes and x-intercepts while vertex form more readily connected to translations of the curves. Students also examined recursive patterns in linear, quadratic, and exponential functions. In this unit they will make connections to these prior experiences as they extend similar reasoning and explorations to power and polynomial functions. Many standards that were introduced in Algebra 1 will be further developed and/or utilized across Algebra 2 units; look for these standards under the Unit Level Standards heading.

Power functions have the general form f(x)= axⁿ. The study of power functions includes exploring the impact of changing the values of the coefficient a and the exponent n. (In this unit, n is kept to positive whole numbers. In unit 4, the study of these models will include values of n that are negative and fractional, as well as whole numbers.) Numeric patterns in tables will be contrasted to linear and quadratic patterns of change and connected to features in the graphs. As n increases students will also note similarities and differences in shapes of the graphs and examine rotational or reflectional symmetry. Attending to the structures in both the graphs and equations will lead to generalizations about power functions with even exponents compared to power functions with odd exponents. End behavior is introduced as a way to compare and contrast the ranges. Students will use multiple representations to first explore and then generalize that the ranges for odd degree power functions are all real numbers and the ranges for even degree power functions is restricted.

Polynomial functions are the sum of power functions and can be represented symbolically by: f(x) = a_nxⁿ+ a_n-1x^n-1+…+ a₁x¹+a₀x⁰, where n is a whole number, and the coefficients, an,are real numbers. Looking for patterns in tables, graphs, and equations will help develop an understanding of characteristics of odd and even degree polynomial functions as an extension to power functions. These patterns show the relationship of the degree of the polynomial function to the possible number of zeroes and the general shape of the graphs. The key features of the graph and the relationship to the equation and table provide the basis for developing understanding of this broader class of polynomial functions.

Students will also perform arithmetic operations on polynomials. They will rewrite polynomials in factored form to reveal the zeroes using clues from the tables and graphs and structures in equations of the functions. They extend their understanding of the relationship between zeroes and factors from quadratics to polynomial functions. They see that for a polynomial p(x), and a number a, ​p(a) = 0 if and only if (x-a) is a factor of p(x). (Division by (x-a) is postponed for unit 4 where the Remainder Theorem will be used.) This unit could also include an introduction to the Binomial Theorem and Pascal's Triangle to demonstrate expansion of (x+y)^​n.

In Algebra 1, students solved systems of linear equations both approximately, with tables and graphs, and exactly using algebraic strategies.They found approximate solutions for systems of equations consisting of both linear and exponential equations. In addition they found both approximate and exact solutions for systems that include linear, exponential and quadratic equations. In this unit, students will extend these concepts to include systems containing polynomial functions.

There are standards listed in this section for two reasons.

1. The standards have been modified to be appropriate for this unit. Text in gray font is part of the Michigan K-12 standard but does not apply to this unit. Text in brackets
   denotes a modification that has been made to the standard.
2. The standards contain content that is developed and/or utilized across multiple units.

Modified For this Unit

NA for this unit

Developed and/or Utilized Across Multiple Units

Seeing Structures in Expressions

HSA-SSE.A. Interpret the structure of expressions.

HSA-SSE.A.1. Interpret expressions that represent a quantity in terms of its context.

HSA-SSE.A.1a. Interpret parts of an expression, such as terms, factors, and coefficients.

HSA-SSE.A.1b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)ⁿ as the product of P and a factor not depending on P.

HSA-SSE.A.2. Use the structure of an expression to identify ways to rewrite it. For example, see x⁴ – y⁴ as (x²)² – (y²)², thus recognizing it as a difference of squares that can be factored as (x² – y²)(x² + y²).

Reasoning with Equations & Inequalities

HSA-REI.D. Represent and solve equations and inequalities graphically.

HSA-REI.D.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

HSA-REI.D.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.

Interpreting Functions

HSF-IF.A. Understand the concept of a function and use function notation.

HSF-IF.A.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

HSF-IF.A.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

HSF-IF.C.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

1. How do changes in the values of the parameters in a polynomial/power function change the behavior of the graph and/or table?
2. How can the patterns in tables, graphs, and equations help determine the characteristics (e.g. shape, number of zeros, end behavior, odd/even) of polynomial functions?
3. What are some similarities and differences between polynomial, quadratic, exponential, and linear functions?
4. How are operations with polynomial functions similar to or different from operations with integers?
5. How is solving a system consisting of a linear or a non linear equation and a polynomial equation similar/different to solving linear systems and/or quadratic equations?