#### 3.1 Polynomial Functions and their graphs class notes

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Transcript 3.1 Polynomial Functions and their graphs class notes

3.1 Polynomial Functions and
their Graphs
Polynomial Functions
• A polynomial function of degree n is a function of
the form
P(x) a n x n a n1x n 1 ... a1x a 0
where n is a nonnegative integer and a n 0
The numbers a0,a1,a2,…,an are called coefficients
of the polynomial.
The number a0 is the constant term.
The number an, the coefficient of the highest
power, is the leading coefficient.
Parent Graphs
yx
y x2
y x3
y x4
y x5
Transformations of Monomials
• Sketch the graphs of the following functions.
P(x) x
3
Q(x) x 2
4
R(x) 2x 5 4
End Behavior of Poynomials
If Polynomial has an odd degree:
If Polynomial has an even degree:
• If leading coefficient is
positive:
• If leading coefficient is
positive:
• If leading coefficient is
negative:
• If leading coefficient is
negative:
Real Zeros of Polynomials
• If P is a polynomial and c is a real number,
then the following are equivalent:
– c is a zero of P.
– x=c is a solution of the equation P(x)=0.
– x-c is a factor of P(x).
– x=c is an x-intercept of the graph of P.
Using Zeros to Graph a Polynomial
Function
• Sketch the graph of the polynomial function:
P(x) x 2 x 1 x 3
Finding Zeros and Graphing a
Polynomial Function
P(x) x3 2x 2 3x
Bump, Wiggle, Cross
x c
• If n is even:
• If n is odd:
n
Finding Zeros and Graphing a
Polynomial Function
P(x) 2x x 3x
4
3
2
Finding Zeros and Graphing a
Polynomial Function
P(x) x 2x 4x 8
3
2
Graphing a Polynomial Function Using
its Zeros
P(x0 x
4
x 2 x 1
3
2
Local Maxima and Minima of
Polynomials
• If P(x) has a degree of n, then the graph of P
has at most n-1 local extrema.
• Determine how many local extrema each
polynomial has.
P1 (x) x 4 x3 16x 2 4x 48
P2 (x) x5 3x 4 5x3 15x 2 4x 15