OpenStax College Algebra Section 5.2 - Study Questions

Explain how to determine the intercepts of a polynomial function.

Given a polynomial function, determine the intercepts. 1. Determine the y-intercept by setting x = 0 and finding the corresponding output value. 2. Determine the x-intercepts by setting f(x) = 0 and solving for the corresponding real input values.

If a polynomial function is in factored form, what would be a good first step in order to determine the degree of the function?

Multiply the actors together to get general form or to find the leading term.

How do you identify the leading term of a polynomial?

The leading term is the term with the highest degree.

How do you use the degree of the polynomial to determine the number of turning points?

The maximum number of turning points is the degree of the polynomial minus 1.

How do you use the degree of the polynomial to determine the number of x-intercepts.

The maximum number of x-intercepts of a polynomial is the degree.

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x → −∞, f(x) → −∞ and as x → ∞, f(x) → −∞.

This end behavior ( down on the left and right ends) would belong to the graph of a polynomial function with even degree and a negative leading coefficient.

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x → −∞, f(x) → ∞ and as x → ∞, f(x) → ∞.

This end behavior (down on the left end and up on the right end) would belong to the graph of a polynomial function with even degree and a positive leading coefficient.

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x → −∞, f(x) → −∞ and as x → ∞, f(x) → ∞.

This end behavior (down on the left end and up on the right end) would belong to the graph of a polynomial function with odd degree and a positive leading coefficient.

What can we conclude if, in general, the graph of a polynomial function exhibits the following end behavior? As x → −∞, f(x) → ∞ and as x → ∞, f(x) → −∞.

This end behavior (up on the left end and down on the right end) would belong to the graph of a polynomial function with odd degree and a negative leading coefficient.

What is the Power Function Form

Where k and p are real numbers.

Power Function

a function that can be represented in this form where k is a constant, the base is a variable, and the exponent, p, is a constant

Polynomial Function

a function that consists of either zero or the sum of a finite number of nonzero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power

Continuous Function

a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph

Smooth Curve

a graph with no sharp corners

Coefficient

a nonzero real number multiplied by a variable raised to an exponent

Imaginary Number

a number in the form bi where

Term of a Polynomial Function

a term of a polynomial function is made up of a numerical coefficient times x raised to a positive integer power

Arrow Notation

a way to represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a variable

Intermediate Value Theorem

for two numbers a and b in the domain of f, if a < b and f (a) ≠ f (b), then the function f takes on every value between f (a) and f (b); specifically, when a polynomial function changes from a negative value to a positive value, the function must cross the x-axis

Zeros

in a given function, the values of x at which y=0, also called roots

End Behavior

the behavior of the graph of a function as the input decreases without bound and increases without bound

Leading Coefficient

the coefficient of the leading term

Degree

the highest power of the variable that occurs in a polynomial

Turning Point

the location at which the graph of a function changes direction

Leading Term

the term containing the highest power of the variable

Describe the graph of a polynomial of nth degree.

A smooth continuous curve with at most n-1 turning points and at most n x-intercepts.