Polynomial Functions and Modeling

Polynomial Functions

A polynomial function can be written as a sum or difference of terms with a variable raised to a nonnegative integer exponent.

A polynomial in one variable is a sum or difference of terms of the form axnax^n, where aa is a real number and nn is a nonnegative integer.

The degree of a term in a polynomial is the exponent of the variable. For example, the term 5x35x^3 has degree 3. A constant term has degree zero. A variable with no exponent has degree 1. When written in standard form, where an0a_n\neq 0, the terms of a polynomial decrease in degree as they are written from left to right:
anxn+an1xn1+an2xn2+...+a1x+a0a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} +...+a_{1}x + a_{0}
The degree of a polynomial is the greatest degree of a term in the polynomial. In standard form, the degree of the polynomial is the degree of the first term. The leading coefficient is the number multiplied by the variable in a polynomial written in standard form, which places the term with the greatest degree first. A polynomial function is any function whose rule is a polynomial in one variable. An example of a polynomial function of degree 3 with a leading coefficient of 2 is:
f(x)=2x3+x26x+1f(x)=2x^3+x^2-6x+1
  • A constant function is a polynomial function of degree zero, such as f(x)=2f(x)=2.
  • A linear function is a polynomial function of degree 1, such as f(x)=3x+2f(x)=-3x+2.
  • A quadratic function is a polynomial function of degree 2, such as f(x)=x23x+2f(x)=x^2-3x+2.
  • A cubic function is a polynomial function of degree 3, such as f(x)=4x3+x23x+2f(x)=4x^3+x^2-3x+2.

Even and Odd Polynomial Functions

A polynomial function can be even, odd, or neither.

Polynomial functions can be even, odd, or neither. The graph of an even function is symmetric with respect to the yy-axis, and the graph of an odd function is symmetric with respect to the origin.

For a polynomial function, if all the terms have even degrees, the function is even. A constant term has a degree of zero, which is even. If all the terms have odd degrees, the function is odd. If there are terms with both even and odd degrees, the function is neither even nor odd. For example:

  • The function f(x)=3x4x2+7f(x)=3x^4-x^2+7 has terms of degree 4, 2, and zero, so the function is even.
  • The function f(x)=2x5+5xf(x)=2x^5+5x has terms of degree 5 and 1, so the function is odd.
  • The function f(x)=x43x2+6x4f(x)=x^4-3x^2+6x-4 has terms of degree 4, 2, 1, and zero, so the function is neither even nor odd.

Knowing that a function is even or odd can help in graphing the function. For a function that is even, if the point (x,y)(x,y) is on the graph, then the point (x,y)(-x,y) is also on the graph. For a function that is odd, if the point (x,y)(x,y) is on the graph, then the point (x,y)(-x,-y) is also on the graph. Therefore, it is only necessary to locate points for positive values of xx, and the points with negative values of xx can be found using symmetry.

Note that a polynomial function can have an even degree and not be an even function or can have an odd degree and not be an odd function. For example, a polynomial has a degree 4, which is even, but the function is neither even nor odd:
f(x)=x43x2+6x4f(x)=x^4-3x^2+6x-4
It is not symmetric with respect to the yy-axis; however, it does have the same end behavior as an even function.

End Behavior of Polynomials

The degree of a polynomial and the sign of the leading coefficient determine the end behavior of the polynomial.
The end behavior of a function describes how the values of a function change as the input values approach positive or negative infinity. The sign of the leading coefficient and the degree of the polynomial determine the end behavior of the polynomial. For example, in a polynomial where an0a_n\neq 0:
f(x)=anxn+an1xn1+an2xn2+...+a1x+a0f(x)=a_{n}x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} +...+a_{1}x + a_{0}
  • When nn is even and an>0a_n>0, the graph approaches \infty as xx approaches both \infty and -\infty.
  • When nn is even and an<0a_n<0, the graph approaches -\infty as xx approaches both \infty and -\infty.
  • When nn is odd and an>0a_n>0, the graph approaches -\infty as xx approaches -\infty and approaches \infty as xx approaches \infty.
  • When nn is odd and an<0a_n<0, the graph approaches \infty as xx approaches -\infty and approaches -\infty as xx approaches \infty.
If the degree of a polynomial function is even, then the end behavior is the same as xx approaches positive or negative infinity. If the degree of a polynomial function is odd, the end behavior is opposite as xx approaches positive or negative infinity.

Extrema of Polynomial Functions

Turning points of a graph may be local or global minima or maxima of the function.

A turning point of a graph is a point where the graph changes from increasing to decreasing or from decreasing to increasing.

  • If the graph changes from increasing to decreasing ( \nearrow\ \searrow), the turning point is a local maximum.
  • If the graph changes from decreasing to increasing ( \searrow\ \nearrow), the turning point is a local minimum.
  • The point where the value of a function is greatest is the global maximum. The maximum value of a function is the greatest value in the range.
  • The point where the value is least is the global minimum. The minimum value of a function is the least value in the range.
  • If the range of a function extends to infinity, it may not necessarily have a global maximum or global minimum.
A polynomial function of degree nn will have at most n1n-1 turning points. For instance:
f(x)=x76x5+3x3+8x+2f(x)=x^7-6x^5+3x^3+8x+2
The polynomial function has degree 7. So, the value of nn is 7. This means that the graph can have up to n1n-1, or 6, turning points. A graph shows that this function has only 4 turning points.

Use the trace feature of a graphing utility to estimate turning points.

  • local maximum: (1.97,26.18)(-1.97,26.18)
  • local minimum: (0.89,4.33)(-0.89,-4.33)
  • local maximum: (0.89,8.33)(0.89,8.33)
  • local minimum: (1.97,22.18)(1.97,-22.18)
The graph shows that the range of the function is all real numbers, so there is no global maximum or minimum. The turning points occur at local extrema (maxima or minima).
The turning points of a graph contribute to determining its local and global maxima and minima. The graph of f(x)=x76x5+3x3+8x+2f(x)=x^7-6x^5+3x^3+8x+2, for example, has 4 turning points: 2 local maxima and 2 local minima. There is no global maximum or minimum.

Writing a Polynomial Function from a Graph

The features of the graph of a polynomial function, including the intercepts and number of turning points, can be used to write the rule of the function.
A zero of a function is a value of xx that makes the value of the function zero. Each zero of a function f(x)f(x) corresponds to an xx-intercept of the graph of the function, which is a value of xx where the graph intersects or touches the xx-axis. These xx-values are also solutions, or roots, of the related equation:
f(x)=0f(x)=0
The zeros of a polynomial function are related to the factored form of the polynomial. If xcx-c is a factor of the polynomial, then cc is a zero of the function. If xcx-c is a factor of a polynomial function, the multiplicity of the zero x=cx=c is the number of times that the factor appears, or the exponent of the factor. For example:
f(x)=x3(x1)(x+2)2f(x)=x^3(x-1)(x+2)^2
The first factor has an exponent of 3, which indicates a zero of multiplicity 3 at x=0x=0. The second factor has an exponent of 1, which indicates a zero of multiplicity 1 at x=1x=1. The third factor has an exponent of 2, which indicates a zero of multiplicity 2 at x=2x=-2. On a graph of a polynomial function, a close-up view of a zero with multiplicity nn looks like a power function of the form:
f(x)=axnf(x)=ax^n
The multiplicity of a zero, or the exponent of the factor in a polynomial function, provides information on how the zero of the function might look like on a graph.
When given the graph of a polynomial function, key features can be used to write an equation of the function. The graph can also give information about end behavior, turning points, zeros, multiplicity, and the yy-intercept.
Step-By-Step Example
Write an Equation by Analyzing a Graph
Use key features of the graph to write an equation of the function. Assume all relevant information is shown in the graph.
Step 1

Use the end behavior to determine whether the degree is odd or even.

As xx approaches -\infty, f(x)f(x) approaches \infty.

As xx approaches \infty, f(x)f(x) approaches \infty.

The end behavior is the same as xx approaches both negative and positive infinity. So, the degree is even.

The function increases as xx approaches positive infinity. So the leading coefficient is positive (a>0)(a>0).

Step 2

Identify turning points.

There are 3 turning points, which indicates that the degree of the polynomial is at least 4 (n4)(n\ge4).

Step 3

Identify zeros, and write factors of the polynomial.

The xx-intercepts of the graph give the zeros of the function: –1, 1, 2.

Use the zeros to write linear factors of the function rule.

The factors are:
(x+1)(x1)(x2)\begin{gathered}(x+1)\\\\(x-1)\\\\(x-2)\end{gathered}
Step 4

Identify the multiplicity of each zero.

Around the zeros at x=1x=-1 and x=1x=1, the graph looks like a linear function. The zeros have multiplicity 1.

Around the zero at x=2x=2, the graph looks like a quadratic function. The zero has multiplicity 2. The factor that corresponds to the zero is squared.

The function rule includes the expression:
(x+1)(x1)(x2)2(x+1)(x-1)(x-2)^2
Step 5

Use the yy-intercept to find the leading coefficient, aa.

The yy-intercept is –4. Substitute the coordinates of the yy-intercept into the equation.
y=a(x+1)(x1)(x2)24=a(0+1)(01)(02)24=a(1)(1)(2)24=a(1)(1)(4)4=a(4)1=a\begin{aligned}y&=a(x+1)(x-1)(x-2)^2\\-4&=a(0+1)(0-1)(0-2)^2\\-4&=a(1)(-1)(-2)^2\\-4&=a(1)(-1)(4)\\-4&=a(-4)\\1&=a\end{aligned}
Solution
The algebraic rule for the function is:
f(x)=(x+1)(x1)(x2)2f(x)=(x+1)(x-1)(x-2)^2
The rule can be verified by using a graphing calculator or graphing utility on a computer.

Solving Polynomial Equations by Graphing

Given the graph of a polynomial, solutions of the related equation can be identified by estimating the xx-intercepts.
Given the graph of a polynomial, solutions of the related equation (the polynomial set equal to zero) can be found by estimating the xx-intercepts.
Step-By-Step Example
Estimating Solutions from a Graph
The graph shows the graph of the function:
f(x)=x43x3f(x)=x^4-3x^3
Use the graph to estimate the solutions of the related equation:
x43x3=0x^4 - 3x^3 = 0
Step 1
Inspect the graph of the function:
f(x)=x43x3f(x)=x^4-3x^3
It crosses the xx-axis at two points, the xx-intercepts. The xx-intercepts represent the solutions of the equation:
x43x3=0x^4 - 3x^3 = 0
From the graph, the solutions of the equation appear to be x=0x=0 and x=3x=3.

The trace feature of a graphing utility can also be used to estimate the xx-intercepts.

Step 2
Substitute the solutions found from the graph into the equation to check for reasonableness.
x43x3=004303=?000=?00=0x43x3=034333=?08181=?00=0\begin{aligned}x^4-3x^3&=0\\0^4-3\cdot0^3&\stackrel{?}{=}0\\0-0&\stackrel{?}{=}0\\0&=0\;\;\checkmark\end{aligned}\;\;\;\;\;\;\;\;\;\;\begin{aligned}x^4-3x^3&=0\\3^4-3\cdot3^3&\stackrel{?}{=}0\\81-81&\stackrel{?}{=}0\\0&=0\;\;\checkmark\end{aligned}
Solution
The solutions of the equation are x=0x=0 and x=3x=3.