Download presentation

Presentation is loading. Please wait.

Published bySonia Walding Modified over 2 years ago

1
3.5 Higher – Degree Polynomial Functions and Graphs

2
Polynomial Function Definition: A polynomial function of degree n in the variable x is a function defined by Where each a i (0 ≤ i ≤ n-1) is a real number, a n ≠ 0, and n is a whole number. What’s the domain of a polynomial function? P(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0

3
Get to know a polynomial function P(x) = a n x n + a n-1 x n-1 + … + a 1 x + a 0 a n : Leading coefficient a n x n : Dominating term a 0 : Constant term

4
Cubic Functions P(x) = ax 3 + bx 2 + cx + d (b)(a) (d)(c)

5
Quartic Functions P(x) = ax 4 + bx 3 + cx 2 + dx + e (b)(a) (d)(c)

6
Extrema Turning points: points where the function changes from increasing to decreasing or vice versa. Local maximum point: the highest point at a peak. The corresponding function values are called local maxima. Local minimum point: the lowest point at a valley. The corresponding function values are called local minima. Extrema: either local maxima or local minima.

7
Absolute and Local Extrema Let c be in the domain of P. Then (a) P(c) is an absolute maximum if P(c) ≥ P(x) for all x in the domain of P. (b) P(c) is an absolute minimum if P(c) ≤ P(x) for all x in the domain of P. (c) P(c) is an local maximum if P(c) ≥ P(x) when x is near c. (d) P(c) is an local minimum if P(c) ≤ P(x) when x is near c.

8
Example Local minimum point Local minimum point Local minimum & Absolute minimum point Local minimum point Local minimum point A function can only have one and only one absolute minimum of maximum

9
Hidden behavior Hidden behavior of a polynomial function is the function behaviors which are not apparent in a particular window of the calculator.

10
Number of Turning Points The number of turning points of the graph of a polynomial function of degree n ≥ 1 is at most n – 1. Example: f(x) = x f(x) = x 2 f(x) = x 3

11
End Behavior Definition: The end behavior of a polynomial function is the increasing of decreasing property of the function when its independent variable reaches to ∞ or - ∞ The end behavior of the graph of a polynomial function is determined by the sign of the leading coefficient and the parity of the degree.

12
End Behavior Odd degree a > 0 a < 0 Even degree a > 0 a < 0

13
example Determining end behavior Given the Polynomial f(x) = x 4 –x 2 +5x -4

14
X – Intercepts (Real Zeros) Theorem: The graph of a polynomial function of degree n will have at most n x-intercepts (real zeros). Example: P(x) = x 3 + 5x 2 +5x -2

15
Comprehensive Graphs A comprehensive graph of a polynomial function will exhibit the following features: 1. all x-intercept (if any) 2. the y-intercept 3. all extreme points(if any) 4. enough of the graph to reveal the correct end behavior

16
example 1. f(x) = 2x 3 – x 2 -2 2. f(x) = -2x 3 - 14x 2 + 2x + 84 a) what is the degree? b) Describe the end behavior of the graph. c) What is the y-intercept? d) Find any local/absolute maximum value(s).... local/absolute maximum points. [repeat for minimums] e) Approximate any values of x for which f(x) = 0

17
Homework PG. 210: 10-50(M5), 60, 63 KEY: 25, 60 Reading: 3.6 Polynomial Fncs (I)

Similar presentations

Presentation is loading. Please wait....

OK

Graphs of Polynomial Functions

Graphs of Polynomial Functions

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on condition monitoring maintenance Ppt on atrial septal defect treatment Ppt on classical economics school Ppt on personality development presentation slides Ppt on sea level rise simulation Ppt on nuclear family and joint family system Ppt on ufo and aliens videos Ppt on standing order form Ppt on marketing class 12 Ppt on 9/11 conspiracy theory facts