Polynomial Functions and Graphs
Higher Degree Polynomial Functions and Graphs
an is called the leading coefficient n is the degree of the polynomial a0 is called the constant term
Polynomial Function
A polynomial function of degree n in the variable x is a function defined by
where each ai is real, an 0, and n is a whole number.01
11)( axaxaxaxP n
nn
n
Polynomial Functions
The largest exponent within the polynomial determines the degree of the polynomial.
Polynomial Function in
General FormDegree Name of
Function
1 Linear2 Quadratic3 Cubic4 Quarticedxcxbxaxy 234
dcxbxaxy 23
cbxaxy 2
baxy
Polynomial Functions
f(x) = 3
ConstantFunction
Degree = 0
Maximum Number of
Zeros: 0
f(x) = x + 2LinearFunction
Degree = 1
Maximum Number of
Zeros: 1
Polynomial Functions
f(x) = x2 + 3x + 2QuadraticFunction
Degree = 2Maximum Number of
Zeros: 2
Polynomial Functions
f(x) = x3 + 4x2 + 2
Cubic Function
Degree = 3
Maximum Number of
Zeros: 3
Polynomial Functions
Quartic Function
Degree = 4
Maximum Number of
Zeros: 4
Polynomial Functions
Leading Coefficient
The leading coefficient is the coefficient of the first term in a polynomial when the terms are written in descending order by degrees.
For example, the quartic function f(x) = -2x4 + x3 – 5x2 – 10 has a leading
coefficient of -2.
The Leading Coefficient Test
As x increases or decreases without bound, the graph of the polynomial function
f (x) = anxn + an-1x
n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)
eventually rises or falls. In particular,
For n odd: an > 0 an < 0
As x increases or decreases without bound, the graph of the polynomial function
f (x) = anxn + an-1x
n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)
eventually rises or falls. In particular,
For n odd: an > 0 an < 0
If the leading coefficient is positive, the graph falls to the left and rises to the right.
If the leading coefficient is negative, the graph rises to the left and falls to the right.
Rises right
Falls left
Falls right
Rises left
As x increases or decreases without bound, the graph of the polynomial function
f (x) = anxn + an-1x
n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)
eventually rises or falls. In particular,
For n even: an > 0 an < 0
As x increases or decreases without bound, the graph of the polynomial function
f (x) = anxn + an-1x
n-1 + an-2xn-2 +…+ a1x + a0 (an ¹ 0)
eventually rises or falls. In particular,
For n even: an > 0 an < 0
If the leading coefficient is positive, the graph rises to the left and to the right.
If the leading coefficient is negative, the graph falls to the left and to the right.
Rises right
Rises left
Falls left
Falls right
The Leading Coefficient Test
Example
Use the Leading Coefficient Test to determine the end behavior of the graph of f (x) = x3 + 3x2 - x - 3.
Falls left
yRises right
x
Determining End Behavior
Match each function with its graph.
4 2
3 2
( ) 5 4
( ) 3 2 4
f x x x x
h x x x x
47)(
43)(7
26
xxxkxxxxg
A. B.
C. D.
Quartic Polynomials
Look at the two graphs and discuss the questions given below.
1. How can you check to see if both graphs are functions?
3. What is the end behavior for each graph?
4. Which graph do you think has a positive leading coeffient? Why?
5. Which graph do you think has a negative leading coefficient? Why?
2. How many x-intercepts do graphs A & B have?
Graph BGraph A-5 -4 -3 -2 -1 1 2 3 4 5
-14
-12
-10
-8
-6
-4
-2
2
4
6
8
10
-5 -4 -3 -2 -1 1 2 3 4 5
-10
-8
-6
-4
-2
2
4
6
8
10
12
14
x-Intercepts (Real Zeros)
Number Of x-Intercepts of a Polynomial Function
A polynomial function of degree n will have a maximum of n x- intercepts (real zeros).
Find all zeros of f (x) = -x4 + 4x3 - 4x2. -x4 + 4x3 - 4x2 = 0 We now have a polynomial
equation. x4 - 4x3 + 4x2 = 0 Multiply both sides by -1. (optional step)
x2(x2 - 4x + 4) = 0 Factor out x2.
x2(x - 2)2 = 0 Factor completely.
x2 = 0 or (x - 2)2 = 0 Set each factor equal to zero.
x = 0 x = 2 Solve for x.
(0,0) (2,0)
Multiplicity and x-Intercepts
If r is a zero of even multiplicity, then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r. Regardless of whether a zero is even or odd, graphs tend to flatten out at zeros with multiplicity greater than one.
Example
Find the x-intercepts and
multiplicity of f(x) =2(x+2)2(x-3) Zeros are at
(-2,0)(3,0)
Extrema
Turning points – where the graph of a function changes from increasing to decreasing or vice versa. The number of turning points of the graph of a polynomial function of degree n 1 is at most n – 1.
Relative maximum point – highest point or “peak” in an interval function values at these points are called local maxima
Relative minimum point – lowest point or “valley” in an interval function values at these points are called local minima
Extrema – plural of extremum, includes all relativel maxima and local minima
Extrema
Number of Relative Extrema
A linear function has degree 1 and no relative extrema.
A quadratic function has degree 2 with one relative extreme point.
A cubic function has degree 3 with at most two relative extrema.
A quartic function has degree 4 with at most three relative extrema.
How does this relate to the number of turning points?
Comprehensive Graphs
The most important features of the graph of a polynomial function are:
1. intercepts,2. extrema,3. end behavior.
A comprehensive graph of a polynomial function will exhibit the following features:
1. all x-intercepts (if any),2. the y-intercept,3. all extreme points (if any),4. enough of the graph to exhibit end
behavior.
