Polynomial Functions Equations and Graphs Characteristics The Factor Theorem The Remainder Theorem http://www.purplemath.com/modules/polyends5.htm 1
A cross-section of a honeycomb has a pattern with one hexagon surrounded by six more hexagons. Surrounding these is a third ring of 12 hexagons, and so on. The quadratic function h (r) that models the total number of hexagons in a honeycomb, where r is the number of rings is h r r r 2 ( ) 3 3 1 This could also be referred to as a Polynomial Function. 2
Identifying Polynomial Functions Any function in the form: f(x) = a n x n + a n-1 x n - 1 + a n - 2 x n -2 + + a 1 x + a 0 where x is a variable the coefficients of the variable, a n, a n - 1. a 0, are real numbers n is a whole number Circle only the polynomial functions. f x x x 2 ( ) 3 2 f x x x 5 2 ( ) 3 6 3 4 3 2 f ( x) x 9x 10x y = 4x -3 + 8x 2 + 3x - 2 y = 2 y 3x2 6x 10 2x Explain why the other functions are not polynomials. 3
f(x) = a n x n + a n-1 x n - 1 + a n - 2 x n -2 + + a 1 x + a 0 f x x x x x 4 3 2 ( ) 5 5 5 6 The degree of the polynomial function is n, the exponent of the greatest power of x. relates to the number of changes of direction on the graph of the function relates to the direction of opening or end behaviour of graph The leading coefficient is a n, the coefficient of the greatest power of x. relates to the direction of opening or end behaviour of graph degree of 4 +1 The constant term is a 0, since x 0. relates to the y-intercept of the graph of the function -6 4
Characteristics of Polynomial Functions Odd Degree Polynomial Functions Degree 1 Linear y = ax + c One direction a > 0 a < 0 Leading coefficient positive Leading coefficient negative End behaviour - + End behaviour + - Constant term c = y-intercept Constant term c = y-intercept x-intercepts: one x-intercepts one Max or Min? Max or Min? 5
Even Degree Polynomial Function Degree 2 Quadratic y = ax 2 + bx + c Two directions a > 0 a < 0 Leading coefficient positive Leading coefficient negative End behaviour + + End behaviour - - Constant term c = y-intercept Constant term c = y-intercept x-intercepts: none, one, two x-intercepts: none, one, two Max or Min? Max or Min? 6
Your Turn: Graph the given polynomial function and identify the characteristics Cubic y = ax 3 + bx 2 + cx + d Degree Odd or Even? y x 3 possible changes in direction a > 0 a < 0 3 2 y x x x 2 2 y x 3 3 2 y x x x 2 2 Leading coefficient Leading coefficient End behaviour Resembles End behaviour Resembles Constant term Domain Range Constant term Domain Range x-intercepts: x-intercepts: Max or Min? Max or Min? 7
Your Turn: Graph the given polynomial function and identify the characteristics Quartic y = ax 4 + bx 3 + cx 2 + dx + e Degree Odd or Even? possible directions a > 0 a < 0 y x 4 4 3 2 y x x x x 5 5 5 6 y x 4 4 3 2 y x x x x 5 5 5 6 Leading coefficient Leading coefficient End behaviour Resembles End behaviour Resembles Constant term Domain Range Constant term Domain Range x-intercepts: Max or Min? x-intercepts: Max or Min? 8
Your Turn: Graph the given polynomial function and identify the characteristics Quintic y = ax 5 + bx 4 + cx 3 + dx 2 + ex + f Degree Odd or Even possible directions a > 0 a < 0 y x 5 5 4 3 2 y x x x x x 3 5 15 4 12 y x 5 5 4 3 2 y x x x x x 3 5 15 4 12 Leading coefficient Leading coefficient End behaviour Resembles End behaviour Resembles Constant term Domain Range Constant term Domain Range x-intercepts: Max or Min? x-intercepts: Max or Min? 9
Use the following information to answer the next four questions Which graph could represent a polynomial whose leading term is 3x 4 +? Which graph could represent a polynomial whose leading term is 3x 4 +? Which graph could represent a polynomial whose leading term is 4x 3 +? Which graph could represent a polynomial whose leading term is -4x 3 +? B D A C 10
Identify the following characteristics for each polynomial function: the type of function and whether it is of even or odd degree the end behaviour of the graph of the function the number of possible x-intercepts whether the function will have a maximum or minimum value the y-intercept g (x) = -x 3 + 8x 2 + 7x - 1 Cubic function, degree 3, odd End behaviour + - At most 3 x-intercepts At least one x-intercept Neither max nor min y-intercept at -1 f (x) = x 4 + x 2 - x + 10 Quartic function, degree 4, even End behaviour + + At most 4 x-intercepts At least no x-intercepts Absolute minimum value y-intercept at 10 11
The height, h, in metres, above the ground of an object dropped from a height of 60 m is related to the length of time, t, in seconds, that the object has been falling. The formula is h = -4.9t 2 + 60. a) What is the degree of this function? b) What are the leading coefficient and constant of this function? c) What does the constant represent? d) What are the restrictions on the domain of the function? e) Describe the end behaviour of the graph of this function. f) Use the formula to determine how long an object will take to hit the ground if it is dropped from a height of 60 m (nearest tenth of a second). 12
Textbook p. 114 117 Low: 1 4 Medium: 5 7, 9 High:10, 12 13