Algebra 2 5.2 Notes 5.2 (Day One) Graphing Polynomial Functions Date: ______What is a Polynomial Function? Learning Target C: I can write a polynomial function in standard form and identify its degree and leading coefficient. We have talked about linear, quadratic, and cubic functions, and all of these are examples of polynomial functions, which are categorized by their degree. A linear function has a degree of 1, a quadratic is of degree 2, and a cubic is a degree 3 polynomial function. Standard form of a Polynomial Function of Degree n
푛 푛−1 2 푝(푥) = 푎푛푥 + 푎푛−1푥 + ⋯ + 푎2푥 + 푎1푥 + 푎0
Where 푎푛, 푎푛−1, … , 푎2, 푎1, 푎푛푑 푎0 are real number coefficients **Terms must be in order by their exponents, starting with the highest and ending with the lowest** Degree of a Polynomial: the ______exponent when the polynomial is written in standard form.
Leading Coefficient: the coefficient of the ______term when the polynomial is written in standard form. Write each polynomial function in standard form. Then, identify its degree and leading coefficient. A. 푓(푥) = 푥3 + 4푥2 − 푥4 + 1 B. 푝(푥) = 푥 + 9푥3 − 2푥 + 6푥2
Intercept Form of a Polynomial Function
푝(푥) = 푎(푥 − 푥1)(푥 − 푥2) … (푥 − 푥푛)
Where 푎, 푥1, 푥2, … , 푎푛푑 푥푛 are real numbers
The polynomial has degree n, where n is the number of variable factors.
Given each function in Intercept form, write it in standard form, and identify the degree and leading coefficient. A. 푓(푥) = 푥2(푥 + 1) B. 푓(푥) = −3푥(푥 − 1)(푥 + 2)2
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Algebra 2 5.2 Notes Investigating the End Behavior of Simple Polynomial Functions Learning Target D: I can determine the end behavior of a polynomial function from its degree and leading coefficient.
Relating End Behavior with Degree
Graph the following functions on a graphing calculator to discover the relationship between the degree of a polynomial with its end behavior.:
푓(푥) = 푥, 푓(푥) = 푥2, 푓(푥) = 푥3, 푓(푥) = 푥4, 푓(푥) = 푥5, 푎푛푑 푓(푥) = 푥6
Relating End Behavior with Leading Coefficient
Compare the graphs of the following functions with the graphs from the functions above to determine the relationship between the leading coefficient of a polynomial and its end behavior:
푓(푥) = −푥, 푓(푥) = −푥2, 푓(푥) = −푥3, 푓(푥) = −푥4, 푓(푥) = −푥5, 푎푛푑 푓(푥) = −푥6
Fill in the table with your findings: Type of Function End Behavior with Positive End Behavior with Negative Leading Coefficient Leading Coefficient Even Degree
Odd Degree
Given each graph, tell whether the degree of the function is even or odd and identify whether the leading coefficient is positive or negative.
A. B. C.
Degree: Degree: Degree: LC: LC: LC: 2
Algebra 2 5.2 Notes Investigating the Turning Points of the Graphs of Polynomial Functions Recall: A turning point is a point where the graph changes from increasing to decreasing or decreasing to increasing. Turning points result in local minimum or local maximum values. Maximum and Minimum Values Global Local The function never takes on a value that is A maximum or minimum within some interval greater than the maximum or less than the around the turning point that does not need to minimum be (but may be) a global maximum or global minimum
Example) Given the graph, determine the number of turning points, the number of global maximum and/or minimum values, and the number of local maximum and/or minimum values that are not global.
For each graph, tell the number of turning points and the number of Global max/min and/or local max/min values that are not global.
A. B. C.
Turning Points: ______Turning Points: ______Turning Points: ______Global Max: ______Global Max: ______Global Max: ______
Local Max: ______Local Max: ______Local Max: ______
Global Min: ______Global Min: ______Global Min: ______
Local Min: ______Local Min: ______Local Min: ______
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Algebra 2 5.2 Notes Finding the Zeros of a Polynomial Function Learning Target E: I can find the zeros of a polynomial function in intercept form. Recall that the zeros of a function give the graph’s ______. Finding the zeros of a polynomial function is easiest when the function is in intercept form. All we need to do is: ______. Find the x-intercepts and state the degree of each polynomial function. (x-intercepts are always written as ordered pairs, (x, 0) )
A. f (x) x(x 3)(x 1) B. f (x) (x 4)2 (x 1)(x 1)
C. f (x) (x 2)2 (x 6) D. 푝(푥) = 푥(푥 + 5)3
5.2 (Day Two) Graphing Polynomial Functions Date: ______Investigating the Behavior of the Graph of a Polynomial Function at Its Zero Values Notice some of the factors in the functions above had exponents other than 1, meaning they occur more than once. The number of times a factor occurs is called its multiplicity. Let’s see how the multiplicity of a factor affects the behavior of the graph at its related x- intercept! Graph each function on a graphing calculator, and sketch them below. Be sure to accurately plot the x-intercepts.
A. B.
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Algebra 2 5.2 Notes
C. f (x) (x 2)2 (x 6) D. 푝(푥) = 푥(푥 + 5)3
Use what you discovered to fill in the table below: Behavior at x-intercepts
Goes Straight Through Tangent to x-axis (“Bounces”) “Squiggles” Through
Sketching the Graph of a Polynomial Function in Intercept Form Learning Target F: I can use end behavior, x-intercepts, and the y-intercept to graph a polynomial function in intercept form. Sketch the graph of each polynomial function. Identify the x- and y- intercepts. A. 푓(푥) = 푥(푥 + 2)(푥 − 3) B. 푓(푥) = −(푥 − 4)(푥 − 1)(푥 + 1)(푥 + 2)
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Algebra 2 5.2 Notes C. 푓(푥) = −푥2(푥 − 4) D. 푓(푥) = (푥 − 2)(푥 + 1)(푥 + 4)2
Writing Polynomial Functions from their Graphs Learning Target G: I can write a polynomial function from its graph. Write the equation of each graph in intercept form, with integer x-intercepts. Assume the leading coefficient, a, is either 1 or -1. A. B.
Identify zeros and multiplicities: Identify zeros and multiplicities:
Degree: Degree: Leading Coefficient: Leading Coefficient: Equation: Equation:
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Algebra 2 5.2 Notes C. D.
Identify zeros and multiplicities: Identify zeros and multiplicities:
Degree: Degree: Leading Coefficient: Leading Coefficient: Equation: Equation:
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