Polynomial Functions and Some Key Characteristics

Section 3.1 Polynomial Functions and Some Key Characteristics

In this section, we’ll learn about the long-term behaviors of polynomial functions and how to identify this, as well as other key attributes of polynomial functions, based on their algebraic formula.

Subsection Textbook Reference

This relates to content in §5.2 of Algebra and Trigonometry 2e

http://tiny.cc/111Z-Textbook

Exercises Preparation Exercises

1.

(a)

Completely factor \(2x^2 + 2x - 12.\)

(b)

Completely factor \(2x^2 + 3x - 2.\)

(c)

Algebraically find and state the \(x\)-intercepts of the graph of \(f(x) = 2x^2 + 2x - 12\text{.}\)

(d)

Algebraically find and state the \(x\)-intercepts of the graph of \(f(x) = 2x^2 + 3x - 2\text{.}\)

(e)

Algebraically find and state the \(y\)-intercept of the graph of \(f(x) = 2x^2 + 3x - 2\text{.}\)

Exercises Practice Exercises

1.

Let \(f(x) = \frac{1}{2} x^4 +3x^3 -16x\text{,}\) which in factored form is \(f(x) = \frac{1}{2}x(x-2)(x+4)^2\text{.}\)

(a)

Describe the end behavior of the graph of \(y=f(x)\text{.}\)

(b)

At most how many turning points does the graph of \(y=f(x)\) have?

(c)

What is the \(y\)-intercept of \(y=f(x)\text{?}\)

(d)

What are the \(x\)-intercept(s) of \(y=f(x)\text{?}\)

2.

Let \(g(x) = 2x^3 - 5x^2 - 4x + 12\text{,}\) which in factored form is \(g(x) = (2x+3)(x-2)^2\text{.}\)

(a)

Describe the end behavior of the graph of \(y=g(x)\text{.}\)

(b)

At most how many turning points does the graph of \(y=g(x)\) have?

(c)

What is the \(y\)-intercept of \(y=g(x)\text{?}\)

(d)

What are the \(x\)-intercept(s) of \(y=g(x)\text{?}\)

Subsection Definitions

Definition 3.1.1. Power Function.

A power function is a function that can be represented in the form

\begin{equation*} f(x) = kx^p \end{equation*}

where \(k\) and \(p\) are real numbers. \(k\) is called the coefficient.

Definition 3.1.2. Polynomial Function.

A polynomial function is of the form

\begin{equation*} p(x) = a_n x^n y^m + a_{n-1} x^{n-1} + ... + a_0 x + a_0 \end{equation*}

where \(a_n, ~a_{n-1}, ~\dots,~ a_1, ~a_0\) are real numbers, \(a_n\neq 0\text{,}\) and \(n\) is a non-negative integer.

The leading term is the highest degree term, \(a_n x^n\text{.}\)
The degree of the polynomial is \(n\text{.}\)
The leading coefficient is the coefficient of the leading term, \(a_n\text{.}\)

Example 3.1.3.

The polynomial \(5x^2-8x+4\) has a leading term of \(5x^2\text{,}\) is second degree polynomial, and has a leading coefficient of \(5\text{.}\)

Definition 3.1.4. \(x\)-intercept or Horizontal Intercept.

A horizontal intercept or \(x\)-intercept of a graph is a point where the graph intersects the horizontal or \(x\)-axis. This occurs when the function has an output value of 0.

Example 3.1.5.

We can find the horizontal intercepts of the graph of \(f(x) = 5x^2-8x+4\) by solving \(f(x)=0\text{.}\)

Definition 3.1.6. \(y\)-intercept or Vertical Intercept.

A vertical intercept or \(y\)-intercept of a graph is a point where the graph intersects the vertical or \(y\)-axis. This occurs when the function has an input value of 0.

Example 3.1.7.

We can find the vertical intercepts of the graph of \(f(x) = 5x^2-8x+4\) by evaluating \(f(0)\text{.}\)

Definition 3.1.8. End Behavior or Long-Term Behavior.

The end behavior or long-term behavior of a polynomial function is determined by its leading term. The long-term behavior of the polynomial function will be consistent with the power function that is the leading term of the polynomial.

Example 3.1.9.

The end behavior of \(f(x) = 5x^4-7x^2+8x-9\) will be the same as the end behavior of \(y=5x^4\text{.}\)

Definition 3.1.10. Turning Point.

A turning point of a polynomial function is a point at which the graph changes from increasing to decreasing or from decreasing to increasing.

Example 3.1.11.

The graph of \(f(x) = 5x^4 - 7x^2 + 8x - 9\) has at most 3 turning points.

Definition 3.1.12. Root or Zero.

A root or zero of a polynomial function is a value \(r\) for which \(f(r)=0\text{.}\) \(r\) is a zero of a polynomial function \(f\) if and only if \((x-r)\) is a factor of \(f\text{.}\)

Example 3.1.13.

If \(7\) is a zero of a polynomial, then \((x-7)\) is a factor of the polynomial.

Example 3.1.14.

If \((x+3)\) is a factor of a polynomial, then \(-3\) is a zero of the polynomial.

Exercises Exit Exercises

1.

(a)

What does the degree of a polynomial function tell you about the graph of the function?

(b)

What does the leading coefficient of a polynomial function tell you about the graph of the function?

(c)

If you know the graph of a polynomial function has 7 turning points, what can you say about the degree of the function?

(d)

What do we know about a polynomial function’s formula if we know the function has the following behavior?

As \(x\rightarrow -\infty\text{,}\) \(f(x)\rightarrow \infty\) and as \(x\rightarrow \infty\text{,}\) \(f(x)\rightarrow -\infty\text{.}\)

2.

State the degree, leading coefficient, long-term behavior, \(y\)-intercept, and the \(x\)-intercepts for function \(f(x) = \frac{1}{3}(x-4)(x+5)^2(x+1)\text{.}\)

Reflection Reflection

1.

On a scale of 1-5, how are you feeling with the concepts related to the graphical behaviors of functions?

Prev Top Next