advanced functions ppt (chapter 1) part ii
TRANSCRIPT
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Advanced
Functions E-PresentationPrepared by:
Tan Yu HangTai Tzu YingWendy Victoria VazTan Hong YeeVoon Khai SamWei Xin
Part I I
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1.3Equations and Graphs of Polynomials
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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) .
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Example 51320 xxxx
30300190150300 234 f
• 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)
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Example 51320 xxxx
321)( 23 xxxxg
-2 is a zero of multiplicity 2 3 is a zero of multiplicity 1
• 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 zeros of the polynomial since it is where it = 0. These are also called the roots of 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?
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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 crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the 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.
221 xxxf
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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.
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Let’s graph 432 xxxxf
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.
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What can we get from the graph ?
1.Degree of the polynomial function
2.Sign of leading coefficient3. End Behavior
4.X and Y intercepts5. Intervals
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1.Degree of the polynomial function
• This is a EVEN root polynomial function.• Even-degree polynomials are either facing
up or down on both ends.
EVEN-DEGREE
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ODD-DEGREE
• Odd-Degree polynomial have a type of graph by which both the end is at the opposite side.
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2.Sign of leading coefficient
POSITIVE COEFFICIENT NEGATIVE COEFFICIENT
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3.End Behavior• Can be determined through the end of the by which it extended from quadrant _ to quadrant _Quadrant 1 Quadrant 2
Quadrant 4 Quadrant 3
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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.
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5.Intervals
interval
Sign of leading coefficient
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.
Choose a number which is bigger/more than 1 and substitute it into the polynomial equation.
For Example: Y=(X+1)1(X-1)
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The End. Hope you enjoyed our Advanced Functions E-Presentation and learnt something!