1.3 EQUATION AND GRAPHS OF POLYNOMIAL FUNCTIONS

1.3 EQUATION AND GRAPHS OF POLYNOMIAL

FUNCTIONS

OBJECTIVES:

• ZEROS (roots) of polynomial functions.

ORDER E.g) f(x) = (x+2) (x-1)2

• If (x-an), then the zeros of orders, is 2 at x= -1 and a double root.

Value of x such that f(x) = 0y-intercept = x = 0x-intercept = y = 0

Zeros(roots)

Order

X-intercept

Leading Term

Leading Coefficient

Degree Term

Examples: f(x) = -4x7 + 5x4 – 2x + 10

Leading term : The term that the variable

has

it’s highest opponent. In this case, the

leading

term is -4x^7.

Leading Coefficient : The coefficient on the

leading term. So, it would be -4.

Degree Term : The variable, which would

be 7.

GRAPHING A POLYNOMIAL FUNCTIONS

Degree

Sign Of leading

Coefficient

Y-intercept

X-intercept

Leading Point (n-1)

Example : (x-1) (x+1)

X < -1 -1 < x < 1 X > 1Positive Negative Positive

EVEN AND ODD FUNCTIONS

• Even Function

• Odd Function

EVEN FUNCTION is when

f(x) = f(-x), for all x.

Symmetry on the y-axis

Called even because…

ODD FUNCTION is when

-f(x) = f(-x), for all x.

Origin Symmetry.

Called odd because…

1.4 : TRANSFORMATIO

N

a is vertical stretch/compression |a| > 1 = stretches |a| < 1= compressesa < 0= flips the graph upside down b= is horizontal stretch/compression

|b| > 1 = compresses |b| < 1 =stretchesb < 0 =flips the graph left-right c is= horizontal shift

c < 0= shifts to the right c > 0= shifts to the left d =is vertical shift d > 0 =shifts upward d < 0 =shifts downward

TRANSFORMMMEE!!!!

All In One ... !You can do all

transformation in one go using this:

Chapter 1 Polynomial Functions

1.1 Power Functions

a = Coefficient (Real numbers)x = Variable n = Degree (must always be a whole number) All polynomial functions can be written in the form of:

Key Features of Graphs

y = xn, n is oddy = xn , n is even

1.2 Characteristics of Polynomial Functions

Finite Differences

Method 1: Pencil & Paper

Method 2: Graphing Calculator

FD = an! E.g. : 2 = a(1!) a = 2

Value of the Leading Coefficient

Key Features of Graphs of Polynomial Functions with Odd Degree

Key Features of Graphs of Polynomial Functions with Even Degree

What is Rate of Change ???

Rate of change is a measure of the change in one quantity (the

dependent variable) with respect to change in another quantity ( the independent variable)

Rate of Change

Average Rate of Change

A change that takes place over an interval.

Instantaneous Rate of Change A change that takes place in an instant.

1.5 Slopes of Secant and

Average Rate of Change Represents the rate of change over a specific

interval . Corresponds to the slope of a secant between

2 points . Average Rate of Change formula: = y = y2-y1 x x2-x1

the slope between 2 points can be calculated by :

1. A table of values 2. An equation .

ExamplesExample 1 :A new antibacterial spray is tested on a bacterial culture. The table shows the population, P, of the bacterial culture t, minutes after the spray is applied. Determine the average rate of change. From the table with the points (0,800) and (7,37): Average rate of change =P = 37-800 = -109 t 7-0

T(min)

P

0 8001 7992 7823

737

4

652

5 5156 3147 37

During the entire 7 minutes , the number of bacteria decreases on average by 109 bacteria per minute.

Example 2 :A football is kicked into the air such that its height ,h, in metres, after t seconds can be modelled by the function h(t) =-4.9 t2 + 14t +1. Determine the average rate of change of the height for the time interval : [0 , 0.5 ]Solution :

Substitute t=0 , h(0)= -4.9(0)2 + 14 (0) + 1=1

Substitute t= 0.5, h(0.5)=-4.9(0.5)2 + 14(0.5) +1 =6.775

Average rate of change = h = 6.775-1 = 11.55 t 0.5-0

The average rate of change of the height of the football from 0s to 0.5s is 11.55m/s.

1.6 Instantaneous Rate of

Change An instantaneous rate of change corresponds to

the slope of a tangent to a point on a curve . An approximate value can be determined by : 1. A graph Draw a tangent line on the graph and estimating the

slope of the tangent of the graph. 2. A table of values Estimating the point and a nearby point in the table 3. An equation Estimating the slope using a very short interval between

the tangent point and a second point found using the equation

Example :

The function shows a ball thrown into the air according to the equation f(x) = -5x2 + 10x ; where x is time (s) and f is height (m) .Find the instantaneous rate of change of the ball at 1.5 seconds in different ways .

Graph method

Table of values method

An equation method

THE END ! ( like finally )