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 n1x 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
yx
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