Graphing Polynomial Functions Graphing Parabolas End-Behavior Definitions and Theorems Vertical and Horizontal Asymptotes Des Cartes’ Rule of Signs copyright © 2013 by Lynda Aguirre1

2 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 Aguirre3 End Behavior 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? 1) Use a graphing calculator to draw each function on graph paper

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 copyright © 2013 by Lynda Aguirre 4 *Note: If the first term in the polynomial is negative, these directions reverse. (up becomes down and vice versa)

copyright © 2013 by Lynda Aguirre5 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 Aguirre6 Definitions Remainder Theorem: p(c) = remainder The remainder = zero, and 2 is a factor of p(x) Use the remainder theorem to find p(2) See notes on how to do synthetic division or use synthetic division to find the remainder which will be equal to p(c). 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)

copyright © 2013 by Lynda Aguirre7 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 Aguirre8 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 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 Aguirre9 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 Aguirre10 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 Aguirre11 Definitions Imaginary zeros are not covered in this course

copyright © 2013 by Lynda Aguirre12 Examples of polynomial functions

1) f(x) = x (x+2) 2 or f(x) = x 3 + 4x 2 + 4x copyright © 2013 by Lynda Aguirre13 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?

2) f(x) = x copyright © 2013 by Lynda Aguirre14

3) f(x) = (x + 4) 2 (x 2 + 1) copyright © 2013 by Lynda Aguirre15 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

Attributes: Multiplicity copyright © 2013 by Lynda Aguirre16 If the factor has a 1 st power, (x – c), the function passes straight through the x-axis at that zero (c).

Polynomial Graphs-Rational Functions (Asymptotes) copyright © 2013 by Lynda Aguirre17 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.

Vertical Asymptote copyright © 2013 by Lynda Aguirre18 3) Draw a vertical line at that x-value 1) Set “bottom” equal to zero 2) Solve for x Vertical Asymptote Vertical Asymptotes exist where the denominator of a fraction is equal to zero

Horizontal Asymptotes copyright © 2013 by Lynda Aguirre19 b) If n = mc) If n > m a) If n < m horizontal asymptote: no horizontal asymptotes (i.e. the x-axis)

Polynomial Graphs-Rational Functions (Asymptotes) copyright © 2013 by Lynda Aguirre20 Other information not included in this course: Oblique Asymptotes: Do polynomial long division and graph the quotient (the answer). It will be a line

DesCartes’ Rule of Signs copyright © 2013 by Lynda Aguirre21 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.

DesCartes’ Rule of Signs copyright © 2013 by Lynda Aguirre22 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 Aguirre23 DesCartes’ Rule of Signs Plugging in (x) gives us the original equation and count sign changes from left to right Number of positive real zeros (to the right of y-axis): 3 or 1 Red arrows indicate sign changes (+ to – and vice versa) Find the positive zeros 1)Plug in (x) to the function p(x) 2) Count sign changes--# of positive zeros (going down by 2)

copyright © 2013 by Lynda Aguirre24 DesCartes’ Rule of Signs Number of negative real zeros (to the left of y-axis): 2 or 0 Red arrows indicate sign changes (+ to – and vice versa) Find the negative zeros Plug in (-x) into the original equation and count sign changes from left to right 2) Count sign changes--# of negative zeros (going down by 2) Figure out the new signs: positive power: negative cancels negative power: negative remains