2.1 Polynomial Functions of Higher Degree
A polynomial function is defined as:
Things we should know:a) The graph is continuousHas no breaks in it
b) The graph is smoothHas no sharp points
c) If and n is even, then the graph is symmetric to y-axis
d) If and n id odd, then the graph is symmetric to the origin
Power Functions
- Functions that are defined as
- If n is even, then the graph represents that of
- As n gets larger, the graph flattens out as the origin
- Example at right is and
- If n is odd, then the graph represents that of
- As n gets larger, the graph flattens out as the origin
- Example at right is and
Leading Coefficient Test
- The leading coefficient test will tell us the end behavior of the graph (left end and right end)
- If n is even and is positive, the graph rises to the left and right
- LEFT:
- RIGHT:
- Example is
- If n is even and is negative, the graph falls to the left and right
- LEFT:
- RIGHT:
- Example is
- If n is odd and is positive, the graph falls to the left and rises to the right
- LEFT:
- RIGHT:
- Example is
- If n is odd and is negative, the graph rises to the left and falls to the right
- LEFT:
- RIGHT:
- Example is
Zeros of Polynomial Functions
Things we should know:a) The degree of the polynomial is n
b) The function has, at most, n real zeros
c) The graph has, at most, turning points
d) is a zero of the function f
e) is a solution of the polynomial equation
f) is a factor of the polynomial
g) is an x-intercept of the graph of f
Example: Find all real zeros of
The degree of the polynomial is 4The highest power is 4
The graph has, at most, 3 turns
The function has, at most, 4 real zeros
Set the equation equal to zero
Factor out GCF of
Factor into monomials
Set each piece to zero and solve
The graph at the right is the actual graph.
Note the zeros.
Repeated Zeros (Multiplicity)
A factor yields a repeated zero of multiplicity k.
1) If k is odd, the graph crosses the x-axis at .
2) If k is even, the graph touches the x-axis (bounces) at .
EXAMPLE: Find a polynomial of degree 4 with given zeros of
Since the zeros are also factors we can write the equation using
- Multiplying this out will result in a polynomial of degree 4
EXAMPLE: Find a polynomial of degree 5 with given zeros of
- Since the degree is one higher than the number of solutions, we need to
increase the multiplicity on one of the factors to get the desired degree
Sketching the Graph of a Polynomial Function
EXAMPLE: Sketch the graph of
- The degree is 3 and the leading coefficient is negative so we know left side goes up
and the right side goes down
- As
- As
- We solve to get the real zeros
Factor out the GCF of -x
Factor remaining trinomial
This tells us we have zeros at
- All the factors have multiplicity of 1 (which is odd) so the graph crosses at each point
- The graph will have at most 2 turns