3. 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) .
4. 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)
5. 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
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.
13. What can we get from the graph ? Degree of the polynomial function Sign of leading coefficient End Behavior X and Y intercepts Intervals
14.
15.
16. 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
17. 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.
18. 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. Sign of leading coefficient Choose 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!
