Did you know???
i.e. y = -2x3 Since -2 is the leading coefficient and is a negative number, hence, the constant finite
difference for this polynomial function will also be a negative constant
- The number of finite differences is the same as the degree of the polynomial function.
i.e. y = -2x3 Since the degree of the polynomial function is 3, hence, this polynomial function has 3
finite difference / hence, this polynomial function reaches its constant at the 3rd finite difference
- Simple??? I say very this is pretty straight forward. What makes it simpler is the existence of a formula for finding finite differences without using a graphing calculator!
Constant Finite Difference = Leading Coefficient x Degree!
C.F.D = a x n!
For example,
A polynomial function has 4 finite difference and its constant is -48
Therefore, n = 4 >>> Remember? The number of finite differences = The degree of a P. function
Using the formula,
C.F.D = a x n!
-48 = a x (4 x 3 x 2 x 1)
a = -2
Don't believe me? Try more questions Here
At first, I didn't quite understand how an even/odd degree polynomial function affected the maximum/minimum points of its graph. But after i read through this table, everything seemed to make MUCH more sense! Hoorayyy!!!
Description | Odd Degree (n) | Even Degree (2n) | ||
Leading Coefficient | + | - | + | - |
General Shape | Refer to picture A | A reflection in the x-axis of its original shape | A reflection in the x-axis of its original shape | Refer to picture B |
Domain | { x ϵ R } | { x ϵ R } | {x ϵ R } | { x ϵ R } |
Range | { y ϵ R } | { y ϵ R } | { y ϵ R | y has a restriction } | { y ϵ R | y has a restriction } |
End Behavior | Extends from quadrant 3 to quadrant 1 | Extends from quadrant 2 to quadrant 4 | Extends from quadrant 2 to quadrant 1 | Extends from quadrant 3 to quadrant 4 |
Number of absolute maximum / minimum points | 0 | 0 | 1 | 1 |
Total number of local maximum / minimum points | Only a maximum of n-1 | Only a maximum of n-1 | Only a maximum of n-1 | Only a maximum of n-1 |
Total number of x-intercepts | Maximum n, minimum 1 | Maximum n, minimum 1 | Maximum n, minimum 0 | Maximum n, minimum 0 |
Note :
1) Both ends of the graph with an odd degree faces an opposite direction (1 north, 1 south). Whereas, the ends of the graph with an even degree faces the same direction (both north / both south).
2) ONLY the even degree has :
- restrictions for its range (y values)
- ONE absolute maximum / minimum point (ABSOLUTE basically means MOST. So, absolute
maximum/ minimum point is the point on the graph with even degree that has the highest or lowest
peak)
3) ALL the types of polynomial graphs can have a maximum number of local maximum / minimum
points of n-1. Nothing more than that!!!
*** By the way, an absolute maximum / minimum point need not be a turning point on a graph. It can also be an end of the graph which has not restrictions***
I hope this piece of information helped you well! It cleared my doubts, totally!
See you soon!
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