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## Advanced functions part iiPresentation Transcript

E-Presentation
Prepared by:
Part II
Tan Yu Hang
Tai Tzu Ying
Wendy Victoria Vaz
Tan Hong Yee
VoonKhai Sam
Wei Xin
• 1.3Equations and Graphs of Polynomials
• Definitions
interval of increase: an interval over the domain of a function where the value of the function is strictly increasing (going from left to right).
interval of decrease: an interval over the domain of a function where the value of the function is strictly decreasing (going from left to right).
odd function: all odd functions have rotational symmetry about the origin and satisfy the equation f (−x) = − f (x) .
even function: all even functions have symmetry about the y-axis and satisfy the equation f (−x) = f (x) .
• Example
x and y intercepts would be useful and we know how to find those. To find the y intercept we put 0 in for x.
To find the x intercept we put 0 in for y.
Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that’s okay)
• Example
We found the x intercept by putting 0 in for f(x) or y (they are the same thing remember). So we call the x intercepts the zerosof the polynomial since it is where it = 0. These are also called the rootsof the polynomial.
Can you find the zeros of the polynomial?
There are repeated factors. (x-1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and solve we get 1. We then say that 1 is a zero of multiplicity 3 (since it showed up as a factor 3 times).
What are the other zeros and their multiplicities?
-2 is a zero of multiplicity 2
3 is a zero of multiplicity 1
• So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crossesthe x axis at this point (odd multiplicities CROSS) or whether it just touchesthe axis and turns and heads back the other way (even multiplicities TOUCH). Let’s try to graph:
What would the left and right hand behavior be?
You don’t need to multiply this out but figure out what the highest power on an x would be if multiplied out. In this case it would be an x3. Notice the negative out in front.
• Steps for Graphing a Polynomial
• Determine left and right hand behavior by looking at the highest power on x and the sign of that term.
• Determine maximum number of turning points in graph by subtracting 1 from the degree.
• Find and plot y intercept by putting 0 in for x
• Find the zeros (x intercepts) by setting polynomial = 0 and solving.
• Determine multiplicity of zeros
• Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.
• Let’s graph
Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.
Here is the actual graph. We did pretty good. If we’d wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maximum and minimum.
• What can we get from the graph ?
Degree of the polynomial function
End Behavior
X and Y intercepts
Intervals
• 1.Degree of the polynomial function
EVEN-DEGREE
• This is a EVEN root polynomial function.
• Even-degree polynomials are either facing up or down on both ends.
• ODD-DEGREE
• Odd-Degree polynomial have a type of graph by which both the end is at the opposite side.
POSITIVE COEFFICIENT
NEGATIVE COEFFICIENT
• 3.End Behavior
Can be determined through the end of the by which it extended from quadrant _ to quadrant _
• 4. X and Y intercepts
If a polynomial function has a factor
(x-a) that is repeated n times, then x=a is a zero of order.
Example:
(x-2)2=0 has a zero of order 2 at x=2.
• 5.Intervals
For Example: Y=(X+1)1(X-1)
interval
X<1
-1<X<1
X>1
Choose a number which is smaller/lesser than -1 and sub into the equation. Then determine whether it is +/-
Choose a number between -1 and +1. then do the same thing again as you did it one the previous column.