Advanced functions part ii
- 1. Advanced FunctionsE-PresentationPrepared by: Part IITan Yu HangTai Tzu YingWendy Victoria VazTan Hong YeeVoonKhai SamWei Xin
- 2. 1.3Equations and Graphs of Polynomials
- 3. Definitionsinterval 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) .
- 4. Examplex 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)
- 5. ExampleWe 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
- 6. 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.
- 7. Steps for Graphing a PolynomialDetermine left and right hand behavior by looking at the highest power on x and the sign of that term.
- 8. Determine maximum number of turning points in graph by subtracting 1 from the degree.
- 9. Find and plot y intercept by putting 0 in for x
- 10. Find the zeros (x intercepts) by setting polynomial = 0 and solving.
- 11. Determine multiplicity of zeros
- 12. 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 graphJoin 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.
- 13. What can we get from the graph ?Degree of the polynomial functionSign of leading coefficientEnd BehaviorX and Y interceptsIntervals
- 14. 1.Degree of the polynomial function EVEN-DEGREEThis is a EVEN root polynomial function.
- 15. Even-degree polynomials are either facing up or down on both ends.ODD-DEGREEOdd-Degree polynomial have a type of graph by which both the end is at the opposite side.2.Sign of leading coefficientPOSITIVE COEFFICIENTNEGATIVE COEFFICIENT
- 16. 3.End BehaviorCan be determined through the end of the by which it extended from quadrant _ to quadrant _Quadrant 1 Quadrant 2 Quadrant 4 Quadrant 3
- 17. 4. X and Y interceptsIf 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.
- 18. 5.Intervals For Example: Y=(X+1)1(X-1)intervalX<1-1<X<1X>1Choose 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.Sign of leading coefficientChoose a number which is bigger/more than 1 and substitute it into the polynomial equation.
- 19. The End. Hope you enjoyed our Advanced Functions E-Presentation and learnt something!