10.2 Power Functions, End Behavior and Multiplicity
The simplest polynomials are the single powers . Their shapes describe what any polynomial does far from the origin, and the multiplicity of each zero describes what the graph does where it meets the -axis.
Power functions
A power function has the form .
and show the shape for all odd ; and show the shape for all even . The blue dashed curve on the graph visualizes .
, odd
, even
Key points
,,
,,
Domain
all real numbers
all real numbers
Range
or
Symmetry
odd: symmetric about the origin
even: symmetric about the -axis
Remark. The higher in , the more dramatic the curve is (the more flat it becomes near the origin).
Example 10.2.1. Graph .
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Key points of , stretched by :
Then the graph is shifted left and down , so the centre point lands at and the graph crosses the -axis at .
At :, giving the point .
Example 10.2.2. Graph .
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Key points of , reflected over the -axis, then shifted right and up :
Example 10.2.3. Sketch ,, and .
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End behavior
The leading term decides what the graph does as .
Example 10.2.4..
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The leading term is , so as : down on the left, up on the right. In the middle is where the things happen.
Example 10.2.5..
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Reorder: . The leading term is , so when and : both ends go down, with the interesting part in the middle.
Note. If a polynomial has degree , then there will be at most turning points (local max/min).
Note.All polynomials create continuous curves.
Multiplicity
A factor can repeat. has roots ,, and is a triple root of the root of has multiplicity.
Example 10.2.6. Find the zeros of and their multiplicities.
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Here .
Factor
Zero
Repeats
Multiplicity
2 times
(even)
1 time
(odd)
4 times
(even)
5 times
(odd)
These are all -intercepts. Adding up the multiplicities, in : multiplying out the leading parts, , so we know the end behavior is that of .
(Aside: the leading is not a zero. Something like can’t be solved by dividing, since that gives , which does not exist.)
All roots/zeros intersect the -axis, but:
If a root has even multiplicity, the graph will bounce // touch, and never cross.
If a root has odd multiplicity, the graph will cross.
For example, at (multiplicity above), look just to either side: , so the graph is below the axis, and , so it is above: crossing.