Graphing Polynomial Functions Graphing Parabolas EndBehavior Definitions and

Graphing Parabolas Using a graphing calculator, enter the following functions and look at the graphs and tables. Determine what part of each equation is creating each change. copyright © 2013 by Lynda Aguirre 2

End Behavior 1) Use a graphing calculator to draw each function on graph paper 2) Describe what happens on the far left and far right hand side of each graph 3) How does changing the sign of the first term affect each graph? copyright © 2013 by Lynda Aguirre 3

End Behavior Describes whether the left and right ends of the polynomial are going up or down. If the first term in the polynomial has a power of: -Even degree: Both ends go up -Odd degree: Left end goes down, right end goes up *Note: If the first term in the polynomial is negative, these directions reverse. (up becomes down and vice versa) copyright © 2013 by Lynda Aguirre 4

Definitions Factor Theorem: If you plug in the value of x or use synthetic division and get a remainder of zero, then “c” is a zero (x-intercept) and the factor is (x – c) and vice versa. . copyright © 2013 by Lynda Aguirre 5

Definitions Remainder Theorem: p(c) = remainder You can find p(c) either by plugging in “c” for all the x’s in the polynomial (as we did in the previous slide) or use synthetic division to find the remainder which will be equal to p(c). Use the remainder theorem to find p(2) See notes on how to do synthetic division The remainder = zero, and 2 is a factor of p(x) copyright © 2013 by Lynda Aguirre 6

Definitions Fundamental Theorem of Algebra (existence thm): The number of zeros (and factors) of a polynomial is equal to the degree of the polynomial Degree of a polynomial = number of zeros The highest power is the degree This polynomial has a degree of 4, so it has 4 zeros In this context, “Zeros” are the x-intercepts copyright © 2013 by Lynda Aguirre 7

Definitions Intermediate Value theorem: Plug in two x-values, if the answers have different signs, then the function (graph) crossed the x-axis somewhere in between those x’s. We think there is a zero at x= -5, so we are going to test it by plugging in values on either side of it to see if they have opposite signs. 30 12 16 At x=-6, the value is negative(under the x-axis) and at x= -4, it is positive (above the x-axis) So this means that the function crosses the x-axis somewhere in between the two numbers (maybe at x=-5, like we thought) copyright © 2013 by Lynda Aguirre 8

Definitions Rational Zeros (or Location Theorem): Find b/c (all possible factors of p(x)), test them with synthetic division to find those that have a remainder of zero-these are the zeros. Upper and Lower Bounds Theorem: All zeros lie in-between these two extremes (x-axis)- Part of “Location Theorem” Notes Bisection Theorem : Second part of location theorem, used to find factors of p(x) that are in between integers (i. e. usually irrational numbers) See notes on this on these procedures on the greenebox website copyright © 2013 by Lynda Aguirre 9

Definitions Linear Factorization Thm: P(x) of degree n has n factors (x – c) Irreducible quadratic factors (i. e. when factors won’t reduce without producing imaginary numbers) This equation can’t be factored into real number factors, it will produce imaginary zeros which we are not covering in this course copyright © 2013 by Lynda Aguirre 10

Definitions Imaginary zeros are not covered in this course copyright © 2013 by Lynda

Examples of polynomial functions copyright © 2013 by Lynda Aguirre 12

1) f(x) = x (x+2)2 or f(x) = x 3 + 4 x 2 + 4 x Where are the x and y-intercepts? What is the end-behavior? The figure cuts through the x-axis at one zero and bounces off the x-axis for other one. Why? copyright © 2013 by Lynda Aguirre 13

2) f(x) = x 2 - 2 copyright © 2013 by Lynda Aguirre 14

3) f(x) = (x + 4)2(x 2+ 1) need to readjust window settings Notice the bounce off of the zero at x= -4 (caused by the 2 nd power) and the bounce in midair caused by the imaginary zeros from the (x 2+1) term copyright © 2013 by Lynda Aguirre 15

Attributes: Multiplicity If the factor has a 1 st power, (x – c), the

$Polynomial Graphs-Rational Functions (Asymptotes) Vertical Asymptotes exist where the denominator of a fraction is$

Polynomial Graphs-Rational Functions (Asymptotes) Vertical Asymptotes exist where the denominator of a fraction is equal to zero Process: 1) Set “bottom” equal to zero 2) Solve for x 3) Draw a vertical line at that x-value. copyright © 2013 by Lynda Aguirre 17

$Vertical Asymptotes exist where the denominator of a fraction is equal to zero 1)$

Vertical Asymptotes exist where the denominator of a fraction is equal to zero 1) Set “bottom” equal to zero 2) Solve for x Vertical Asymptote 3) Draw a vertical line at that x-value copyright © 2013 by Lynda Aguirre 18

Horizontal Asymptotes a) If n < m b) If n = m c) If

Polynomial Graphs-Rational Functions (Asymptotes) Other information not included in this course: Oblique Asymptotes: Do polynomial long division and graph the quotient (the answer). It will be a line copyright © 2013 by Lynda Aguirre 20

Des. Cartes’ Rule of Signs Use: To determine how many real zeros (x-intercepts) exist in a function and how many are on each side (positive or negative) of the origin. Note: If some are imaginary zeros (they occur in pairs), this takes away 2 zeros at a time. copyright © 2013 by Lynda Aguirre 21

Des. Cartes’ Rule of Signs Steps: Find the positive zeros (to the right of the y-axis) 1) Plug in (x) to the function p(x); 2) Count sign changes--# of positive zeros (going down by 2) Find the negative zeros (to the left of the y-axis) 3) Plug in (-x) to the function p(x); 4) count sign changes-# of negative zeros (going down by 2) copyright © 2013 by Lynda Aguirre 22

Des. Cartes’ Rule of Signs Find the positive zeros 1) Plug in (x) to the function p(x) • Plugging in (x) gives us the original equation and count sign changes from left to right Red arrows indicate sign changes (+ to – and vice versa) 2) Count sign changes--# of positive zeros (going down by 2) Number of positive real zeros (to the right of y-axis): 3 or 1 copyright © 2013 by Lynda Aguirre 23

Des. Cartes’ Rule of Signs Find the negative zeros Plug in (-x) into the original equation and count sign changes from left to right Figure out the new signs: positive power: negative cancels negative power: negative remains Red arrows indicate sign changes (+ to – and vice versa) 2) Count sign changes--# of negative zeros (going down by 2) Number of negative real zeros (to the left of y-axis): 2 or 0 copyright © 2013 by Lynda Aguirre 24