UNIT 8 STUDY SHEET – POLYNOMIAL FUNCTIONS

KEY FEATURES OF POLYNOMIALS Intercepts of a function

o x-intercepts - a point on the graph where y is zero {Also called the zeros of the function.}

o y-intercepts - a point on the graph where x is zero

Relative max/min

o Relative maximum - The point(s) on the graph, which have maximum y- values or second

coordinates “relative” to the points close to them on the graph.

o Relative minimum - The point(s) on the graph, which have minimum y-values or second

coordinates “relative” to the points close to them on the graph.

o These relative extrema (turning points) help determine the intervals for which a function

is increasing or decreasing

Increasing/Decreasing Functions

o A function is said to be increasing when the y-value increases as the x-value increases.

o A function is said to be decreasing when the y-value decreases as the x-value increases.

o The graph must be read from left-to-right to determine if it is increasing or decreasing.

Increasing:

Decreasing:

DEGREE OF A POLYNOMIAL AND END BEHAVIOR End Behavior (Let 𝑓 be a function whose domain and range are subsets of the real numbers. The end behavior of a function 𝑓 is a description of what happens to the y-values of the function

o as approaches positive infinity, and o as approaches negative infinity.

Sketching Graphs of Polynomials in Factored Form

By looking at the factored form of a polynomial, we can identify important characteristics of the

graph such as -intercepts and degree of the function, which in turn allow us to develop a sketch of

the graph.

A polynomial function of degree may have up to -intercepts.

A polynomial function of degree may have up to relative maxima and minima.

How to Graph Polynomials in Factored Form

1. Zeros – Factor the polynomial to find all its real zeros; these are the x-intercepts of the graph.

2. Test Points – Test a point between the x-intercepts to determine whether the graph of the

polynomial lies above or below the x-axis on the intervals determined by the zeros.

3. End Behavior – Determine the end behavior of the polynomial by looking at the degree of the

polynomial and the sign of the leading coefficient.

4. Graph – Plot the intercepts and other points you found when testing. Sketch a smooth curve that

passes through these points and exhibits the required end behavior.

ODD AND EVEN FUNCTIONS EVEN FUNCTIONS

An EVEN function has the following properties: I. Its graph is symmetric about the y-axis (reflection over the y-axis)

II. 𝑓( ) 𝑓( ) Trick: Can you fold the graph in half along the y-axis and it aligns perfectly? Then it is even.

ODD FUNCTIONS

An ODD function has the following properties: I. Its graph is symmetric about the origin (reflection over the point (0, 0)

II. 𝑓( ) 𝑓( ) Trick: Can you rotate the graph upside down and it still looks the same? Then it is odd.

To determine if a function is odd or even when looking at the graph, look for

symmetery!

If symmetric about the y-axis, it is even

If symmetric about the origin, it is odd

ODD AND EVEN FUNCTIONS (Continued)

To determine if a function is odd or even algebraically, substitute into the

given function!

If ( ) ( ), it is even

If ( ) ( ), it is odd

To determine algebraically if a fraction is an odd or even function, you must

factor out a -1 in the numerator and denominator (if possible).

For example:

Synthetic Division, Remainder Theorem, Factor Theorem:

Using Synthetic Division to Find all Factors and Zeros

Steps:

1. Determine a factor of the polynomial (from the graph or given in the question) and prove

algebraically it is a factor.

2. Use synthetic division to find a quotient that is also a factor of the original polynomial.

3. Use this new polynomial (quotient) to find the remaining zeros by using an appropriate method!

1. ( ) ;

x +3 is a factor of ( )

2. ( ) ; ( )

Zeros and Multiplicity of Polynomials

The real zeros of a polynomial function may be found by factoring (where possible) or by finding where the graph touches the x-axis.

The number of times a zero occurs is called its multiplicity. In an equation, the multiplicity of the zero is represented by the exponent of each factor.

The multiplicity of a zero tells you how the graph will interact with the x-axis at a specific

zero. Odd Multiplicity

The graph crosses the x-axis at the given zero. Example: Below is the graph of ( ) ( ) . You can see the graph crosses the x-axis at the zero x = 3.

Even Multiplicity The graph is tangent to the x-axis at the given

zero. Example: Below is the graph of ( ) ( ) . You can see the graph is tangent to the x-axis at the zero x = 3.

Examples: Find the zeros of the following polynomial functions, with their multiplicities.

a. ( ) ( )( )( )

with multiplicity

with multiplicity

b. ( ) ( ) ( )

with multiplicity

with multiplicity

c. ( ) ( )

with multiplicity

USING ZEROS TO WRITE POLYNOMIAL EQUATIONS

We will use the following steps to write a polynomial function from its given zeros:

1. Convert the zeros to factors.

2. Multiply the factors.

3. Combine like terms and write with powers of x in descending order, which is the standard form of

a polynomial function.

REMEMBER: If one zero is irrational or imaginary, another zero must be its conjugate!

For Example: If a given zero is , then another zero must be

For Example: If a given zero is √ , then another zero must be √

WRITING POLYNOMIAL EQUATIONS TO MODEL A GIVEN SITUATION

Steps:

Find the x-intercepts

Write the equation in factored form using the x-intercepts and a constant factor c.

Plug a point from the graph into the equation to find the value of c.

Write equation with c plugged back in to the factored form.