1.3 EQUATION AND GRAPHS OF POLYNOMIAL FUNCTIONS
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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 )