Polynomial Functions and Models
This is my guide on polynomial functions and models.
Polynomial Functions
We can write linear functions as \(f(x)=mx+b\) and quadratic functions as
\(f(x)=ax^2+bx+c\). Each of these functions are polynomial functions. The
domain of a polynomial function is the set of all real numbers.
A polynomial function is a function whose rule is given by a polynomial in one
variable. The degree of a polynomial function is the largest power of x that
appears. The zero polynomial function \(f(x)=0\) is not assigned a degree.
Polynomial functions are among the simplest expressions in algebra. They are
easy to evaluate: only addition and multiplication are required. Because of
this, they are often used to approximate other, more complicated functions.
Example 1
Identifying polynomial functions
Determine which of the following are polynomial functions. For those that are,
state the degree.
1. \(f(x)=2-3x^4\)
2. \(g(x)=\sqrt{x}\)
3. \(h(x)=\frac{x^2-2}{x^3-1}\)
4. \(f(x)=0\)
5. \(g(x)=8\)
6. \(h(x)=-2x^3(x-1)^2\)
Solution:
1. polynomial function of degree 4
2. not a polynomial function because the power is 1/2 which is not an integer
3. not a polynomial function. it is the ratio of two polynomials and the
polynomial in the denominator is of positive degree.
4. polynomial function of degree zero, it is not assigned a degree
5. nonzero constant function, which is a polynomial function of degree 0
6. polynomial function of degree 5
You will learn that the graph of every polynomial function is both smooth and
continuous. By smooth, we mean that the graph contains no sharp corners and by
continuous, we mean that the graph has no gaps or holes and can be drawn without
lifting pencil from paper.
Power Functions
We begin the analysis of the graph of a polynomial function by discussing power
functions, a special kind of polynomial function. A power function of degree n
is a monomial of the form \(f(x)=ax^n\) where a is a real number, not zero, and
n>0.
Examples of power functions include:
\(f(x)=3x\) is degree 1
\(f(x)=-5x^2\) is degree 2
\(f(x)=8x^3\) is degree 3
\(f(x)=-5x^4\) is degree 4
The graph of a power function of degree 1, \(f(x)=ax\), is a straight line, with
slope a, that passes through the origin. The graph of a power function of degree
2, \(f(x)=ax^2\), is a parabola, with vertex at the origin, that opens up if a>0
and down if a<0.
If we know how to graph a power function of the form \(f(x)=x^n\), a compression
or stretch, and a reflection about the x-axis will enable us to obtain the graph
of \(g(x)=ax^n\). Consequently, we shall concentrate on graphing power functions
of the form \(f(x)=x^n\).
We begin with power functions of even degree of the form \(f(x)=x^n,n\geq2\) and
n even. The domain of f is the set of all real numbers, and the range is the set
of nonnegative real numbers. Such a power function is an even function, so its
graph is symmetric with respect to the y-axis. Its graph always contains the
origin and the points (-1,1) and (1,1).
If n=2, the graph is the familiar parabola \(y=x^2\) that opens up, with vertex
at the origin. If \(n\geq4\), the graph of \(f(x)=x^n\), n even, will be closer
to the x-axis than the parabola \(y=x^2\) if \(-1<x<1\) and farther from the
x-axis than the parabola \(y=x^2\) if \(x<-1\) or if \(x>1\).
Properties of Power Functions
1. f is an even function, so its graph is symmetric with respect with respect to
the y-axis.
2. The domain is the set of all real numbers. The range is the set of
nonnegative real numbers.
3. The graph always contains the points (-1,1) and (1,1).
4. As the exponent n increases in magnitude, the graph becomes more vertical
when x<-1 or x>1. But for x near the origin, the graph tends to flatten out and lie closer to the x-axis.
Now we consider power functions of odd degree of the form \(f(x)=x^n,n\geq3\)
and n odd. The domain and range of f are the set of real numbers. Such a power
function is an odd function, so its graph is symmetric with respect to the
origin. Its graph always contains the points (-1,1) and (1,1).
The graph of \(f(x)=x^n\) when n=3 has been shown several times. If \(n\geq5\),
the graph of \(f(x)=x^n\), n odd, will be closer to the x-axis than that of
\(y=x^3\) if -1<x<1 and farther from the x-axis than that of \(y=x^3\) if x<-1
or if x>1.
Properties of Power Functions when x is Odd
1. f is an odd function, so its graph is symmetric with respect to the origin
2. The domain and the range are the set of all real numbers.
3. The graph always contains the points (-1,-1) and (1,1).
4. As the exponent n increases in magnitude, the graph becomes more vertical
when x<-1 or x>1 but for x near the origin, the graph tends to flatten out
and lie closer to the x-axis.
Graph Polynomial Functions using Transformation
The methods of shifting, compression, stretching, and reflection will enable us
to graph polynomial functions that are transformations of power functions.
Example 2
Graph \(f(x)=1-x^5\)
Solution:
First, start with the usual function of \(y=x^5\)
Multiply by -1 to reflect across the x-axis.
This gives us \(y=-x^5\).
Now, add 1 to equation which will shift the graph up 1 unit.
\(y=-x^5+1\) which is the same as:
\(y=1-x^5\)
Example 3
Graph \(f(x)=\frac{1}{2}(x-1)^4\)
Solution:
Take it step by step:
Start with the base function of \(y=x^4\)
Replace x by x-1 which will shift it right by 1 unit.
This gives us \(y=(x-1)^4\).
Now, multiple by 1/2 which gives the graph a compression by a factor of 1/2.
We now have \(y=\frac{1}{2}(x-1)^4\)
Finding the Real Zeros of a Polynomial Function
The previous graph and problem shows a polynomial function with four
x-intercepts. Notice that at the x-intercepts the graph must either cross the
x-axis or touch the x-axis. Consequently, between consecutive x-intercepts the
graph is either above the x-axis or below the x-axis. We will make use of this
property of the graph of a polynomial function.
If a polynomial function f is factored completely, it is easy to solve the
equation \(f(x)=0\) using the zero-product property and locate the x-intercepts
of the graph. For example, if \(f(x)=(x-1)^2(x+3)\), then the solution of the
equation becomes:
\(f(x)=(x-1)^2(x+3)=0\) which factored give us: 1 and -3 as intercepts.
Based on this we can make the following observations.
If f is a function and r is a real number for which f(r)-0, then r is called a
real zero of f.
As a consequence, the following statements are equivalent.
1. r is a real zero of a polynomial function f
2. r is an x-intercept of the graph of f
3. x-r is a factor of f
So, the real zeros of a polynomial function are the x-intercepts of its graph
and they are found by solving the equation \(f(x)=0\).
Example 4
Finding a Polynomial from its Zeros
Find a polynomial of degree 3 whose zeros are -3, 2, and 5.
Solution:
If r is a real zero of a polynomial f, then x-r is a factor of f. This means
that x-(-3)=x+3, x-2, and x-5 are factors of f. As a result, any polynomial of
the form \(f(x)=a(x+3)(x-2)(x-5)\) where a is any nonzero real number,
qualifies. The value of a causes a stretch, compression, or reflection, but does
not affect the x-intercepts.
If the same factor x-r occurs more than once, r is called a repeated or multiple
zero of f. More precisely, we have the following definition.
If (x-r)^m is a factor of a polynomial f and (x-r)^{m+1} is not a factor of f,
then r is called a zero of multiplicity m of f.
Example 5
Identifying Zeros and their Multiplicities
For the polynomial \(f(x)=5(x-2)(x+3)^2(x-1/2)^4\)
Solution:
2 is a zero of multiplicity 1 because the exponent on the factor x-2 is 1
-3 is a zero of multiplicity 2 because the exponent on the factor x+3 is 2
1/2 is a zero of multiplicity 4 because the exponent on the factor x-1/2 is 4
Suppose that it is possible to factor completely a polynomial function and
locate all the x-intercepts of its graph(the real zeros of the function). These
x-intercepts then divide the x-axis into open intervals and, on each such
interval, the graph of the polynomial will be either above or below the x-axis.
Example 6
Graphing a Polynomial Using its x-intercepts
For the polynomial \(f(x)=x^2(x-2)\)
1. Find the x and y intercepts of the graph of f
2. Use the x-intercepts to find the intervals on which the graph of f is above
the x-axis and the intervals on which the graph of f is below the x-axis.
3. Locate other points on the graph and connect all the points plotted with a
smooth continuous curve.
Solution:
1. The y-intercept is \(f(0)=0^2(0-2)=0\). The x-intercepts satisfy the
equation \(f(x)=x^2(x-2)=0\) from which we find \(x^2=0\) or \(x-2=0\) and
\(x=0\) or \(x=2\). The x-intercepts are 0 and 2
2. The two x-intercepts divide the x-axis into 3 intervals.
\((-\infty,0)(0,2)(2,\infty)\). Since the graph of f crosses or touches the x-axis
only at x=0 and x=2, it follows that the graph of f is either above the x-axis
f(x)>0 or below the x-axis f(x)<0 on each of these 3 intervals. To see where the
graph lies, we only need to pick a number in each interval, evaluate f there,
and see whether the value is positive(above the x-axis) or negative(below the
x-axis).
3. in constructing a table, we obtained three additional points on the
graph:(-1,-3)(1,-1)(3,9).
Since the graph of \(f(x)=x^2(x-2)\) is below the x-axis on both sides of 0, the
graph of f touches the x-axis at x=0, a zero of multiplicity 2. Since the graph
of f is below the x-axis for x<2 and above the graph for x>2, the graph of f
crosses the x-axis at x=2, a multiplicity of 1.
If r is a zero of even multiplicity the sign of f(x) does not change from one side to
the other side of r. The graph of f touches the x-axis at r.
If r is a zero of odd multiplicity, the sign of f(x) changes from one side to
the other side of r. The graph of f crosses the x-axis at r.
Behavior Near a Zero
We have just learned how the multiplicity of a zero can be used to determine
whether the graph of a function touches or crosses the x-axis at the zero.
However, we can learn more about the behavior of the graph near its zeros than
just whether the graph crosses or touches the x-axis.
Consider the function\(f(x)=x^2(x-2)\). The zeros of f are 0 and 2. We can see
that the points on the graph of \(f(x)=x^2(x-2)\) and the points on the graph of
\(y=-2x^2\) are indistinguishable near x=0. So, \(y=-2x^2\) describes the
behavior of the graph of \(f(x)=x^2(x-2)\) near x=0.
But how did we know that the function \(f(x)=x^2(x-2)\) behaves like \(y=-2x^2\)
when x is close to 0? Where did \(y=-2x^2\) comes form? Because the zero, 0,
comes from the factor \(x^2\), we evaluate all factors in the function f at 0
with the exception of \(x^2\).
\(f(x)=x^2(x-2)\)
The factor \(x^2\) gives rise to the zero, so we keep the factor \(x^2\) and let
x=0 in the remaining factors to find the behavior near 0.
\(f(x)=x^2(0-2) = -2x^2\)
This tells us that the graph of \(f(x)=x^2(x-2)\) will behave like the graph of
\(y=-2x^2\) near x=0. Now let us discuss the behavior of \(f(x)=x^2(x-2)\) near
x=2, the other zero. Because the zero, 2, comes from the factor x-2, we
evaluate all factors of the function f at 2, with the exception of x-2.
\(f(x)=x^2(x-2) = 2^2(x-2) = 4(x-2)\)
The factor x-2 gives rise to the zero, so we keep the factor x-2 and let z=2 in
the remaining factors to find the behavior near 2.
So the graph of \(f(x)=x^2(x-2)\) will behave like the graph of \(y=4(x-2)\)
near x=2. We can see that the points on the graph of \(f(x)=x^2(x-2)\) and the
points on the graph of \(y=4(x-2)\) are indistinguishable near x=2. So
\(y=4(x-2)\), a line with slope 4, describes the behavior of the graph of
\(f(x)=x^2(x-2)\) near x=2.
By determining the multiplicity of a real zero, we determine that the graph
crosses or touches the x-axis at the zero. By determining the behavior of the
graph near the real zero, we determine how the graph touches or crosses the
x-axis.
Turning Points
When looking at the previous graph, we cannot be sure how low the graph actually
goes between x=0 and x=2. but we do know that somewhere in the interval (0,2)
the graph of f must change direction (from decreasing to increasing). The points
at which a graph changes direction are called turning points. In calculus, such
points are called local maxima and local minima, and techniques for locating
them are given. So we shall not ask for the location of turning points in our
graphs. Instead, we will use the following result from calculus, which tells us
the maximum number of turning points that the graph of a polynomial function can
have.
If f is a polynomial function of degree n, then f has at most n-1 turning
points.
If the graph of a polynomial function f has n-1 turning points, the degree of f
is at least n.
For example, the graph of \(f(x)=x^2(x-2)\) is the graph of a polynomial of degree
3 and 3-1=2 turning points: one at (0,0) and the other somewhere between x=0 and
x=2.
Example 7
Identifying the graph of a polynomial function
Which of the graphs could be the graph of a polynomial function? For those that
could, list the real zeros and state the least degree the polynomial can have.
For those that could not, say why not.
Solution:
1. This cannot be the graph of a polynomial function because of the gap that
occurs at x=-1. Remember, the graph of a polynomial function is continuous with
no gaps or holes.
2. This could be the graph of a polynomial function. It has 3 real zeros at -2,
1, and 2. Since the graph has 2 turning points, the degree of the polynomial
function must be at least 3.
3. This cannot be the graph of a polynomial function because of the cusp at x=1.
Remember, the graph of a polynomial function is smooth.
4. This could be the graph of a polynomial function. It has 2 real zeros at -2
and 1. Since the graph has 3 turning points, the degree of the polynomial
function is at least 4.
For very large values of x, either positive or negative, the graph of
\(f(x)=x^2(x-2)\) looks like the graph of \(y-=x^3\). To see why, we write f in
the form:
\(f(x)=x^2(x-2)=x^3-2x^2=x^3(1-\frac{2}{x})\)
Now, for large values of x, either positive or negative, the term 2/x is close to
0, so for large values of x:
\(f(x)=x^3-2x^2=x^3(1-\frac{2}{x})=x^3\)
The behavior of the graph of a function for large values of x, either positive
or negative, is referred to as its end behavior.
Example 8
Identifying the graph of a polynomial function
Which of the graphs could be the graph of:
\(f(x)=x^4+5x^3+5x^2-5x-6\)
Solution:
The y-intercept of f is f(0)=-6 so we can eliminate any graph whose y-intercept
is positive. Being positive means the curve crosses the y-axis above 0.
We don't have any methods for finding the x-intercept of f, so we move on to
investigate the turning points of each graph. since f is of degree 4, the graph
of f has at most 3 turning points. So, we eliminate the graph that has 5 turning
points.
Now, we look at end behavior. For large values of x, the graph of f will behave
like the graph of \(y=x^4\). This eliminates the graph whose end behavior is
like the graph of \(y=-x^4\).
Only the graph left is the graph which has 3 turning points and behaves like
\(x^4\).
Summary of Polynomial Functions
Degree of polynomial f: n
Maximum number of turning points: n-1
At a zero of even multiplicity: the graph of f touches the x axis
At a zero of odd multiplicity: the graph of f crosses the x-axis.
Between zeros, the graph of f is either above or below the x-axis
End behavior: For large |x|, the graph of f behaves like the graph of of
\(y=a_{n}x^{n}\).
Example 9
Analyzing the Graph of a Polynomial Function
For the polynomial \(f(x)=x^3+x^2-12x\) :
1. Find the x and y-intercepts of the graph of f
2. Determine whether the graph crosses or touches the x-axis at each
x-intercept
3. End behavior: Find the power function that the graph of f resembles for large
values of |x|
4. Determine the maximum number of turning points on the graph of f
5. Determine the behavior of the graph of f near each x-intercept
6. Put all the information together to obtain the graph of f
Solution:
1. The y-intercept is 0 because \(f(0)=0^3 +0^2-12(0)\)
To find the x-intercepts, if any, we first factor f
\(x(x^2+x-12 = x(x+4)(x-3)\) so x=0, -4, and 3
2. Since each real zero is of multiplicity 1, the graph of f will cross the
x-axis at each x-intercept
3. End behavior: the graph of f resembles that of the power function \(y=x^3\)
for large values of |x|
4. The graph of f will contain at most two turning points because \(x^3\) is the
highest exponent
5. The three x-intercepts are -4, 0, and 3
Near -4: \(-4(x+4)(-4-3) = 28(x+4)\) A line with slope 28
Near 0: \(x(0+4)(0-3)=-12x\) A line with slope -12
Near 3: \(3(3+4)(x-3)=21(x-3)\) A line with slope 21
6. Put all of this together on a piece of paper then use a calculator to graph
and see if it is close.
Example 10
Analyzing the Graph of a Polynomial Function
\(f(x)=x^2(x-4)(x+1)\)
Solution:
1. The y-intercept is 0 because f(0)=0
2. The x-intercepts are 0, 4, and -1
The intercept 0 is a multiplicity of 2 so the graph of f will touch the
x-axis 4 and -1 are zeros of multiplicity 1, so the graph of f will cross the x-axis at 4 and -1.
3. End behavior: the graph of f resembles that of the power function \(y=x^4\)
for large values of |x|
4. The graph of f will contain at most three turning points because of \(x^4\)
5. The three x-intercepts are -1, 0, and 4
Near -1: \(-1^2(-1-4)(x+1)=-5(x+1)\) A line with slope -5
Near 0: \(x^2(0-4)(0+1)=-4x^2\) A parabola opening down
Near 4: \(4^2(x-4)(4+1)=80(x-4)\) A line with slope 80
6. Graph it on paper then with a calculator
Summary for Analyzing the Graph of a Polynomial
1. Find the y-intercept by letting x=0 and finding the value of f(0)
Find the x-intercepts, if any, by solving the equation f(x)=0
2. Determine whether the graph of f crosses or touches the x-axis at each
x-intercept
3. End behavior: find the power function that the graph of f resembles for large
values of |x|
4. Determine the maximum number of turning points on the graph of f
5. Determine the behavior of the graph of f near each x-intercept
6. Put all the information together to obtain the graph of f
For polynomial functions that have non-integer coefficients and for polynomials
that are not easily factored, we utilize a graphing utility early in the
analysis of the graph. This is because the amount of information that can be
obtained from algebraic analysis is limited.