swbat: −match functions to their parent graphs −find domain and range of functions from a graph...

SWBAT:

−Match functions to their parent graphs

−Find domain and range of functions from a graph

−Determine if a function is even or odd

−Give the domain and range of functions in various notations

Parent Graphs, Domain,

and Range

Domain

All the possible input values (x values) of a function.

Generally the domain of functions is all real numbers.

When is it not?

You will learn about a few more this year.

Range

All the possible output values (y values) of a function.

Each function has it’s own range, and often the best way to determine a range is to look at a graph.

Look at each y-value and see if the graph hits that value at all. If it never does (and never will) then that value is not in the range.

What functions do you know that have limited ranges?

Notation for Domain and Range

Three different notations : Set Notation, Algebraic Notation, and Interval Notation

Set Notation is usually for discrete data (individual points)

Algebraic Notation and Interval Notation are used for continuous data

Set Notation

This table shows us a set of discrete data. We only know about the points given to us.

We need to use Set Notation and list the individual values in the domain and in the range.

Set notation uses brackets (the squiggly ones { } ) with the numbers listed in ascending order and are separated by commas.

Algebraic Notation vs. Interval Notation : Mostly a matter of preference

Algebraic Notation Uses equality and

inequality symbols and variables to define domain and range.

Interval Notation For each continuous section of the graph, write the

starting and ending points inside commas or square brackets, separated by commas.

Parenthesis are used when the values is NOT included in the domain/range

Square brackets [ ] are used when the value IS included in the domain/range

Piecewise

Domain:

Range:

Even/Odd?:

Constant Function

Domain:

Range:

Even/Odd?:

Linear Function

Domain:

Range:

Even/Odd?:

Absolute Value

Domain:

Range:

Even/Odd?:

Quadratic

Domain:

Range:

Even/Odd?:

Square Root

Domain:

Range:

Even/Odd?:

Cubic

Domain:

Range:

Even/Odd?:

Cube Root

Domain:

Range:

Even/Odd?:

Rational

Domain:

Range:

Even/Odd?:

Rational

Domain:

Range:

Even/Odd?:

Logarithmic

Domain:

Range:

Even/Odd?:

Exponential

Domain:

Range:

Even/Odd?:

2.2 Polynomial Functions of Higher Degree

SWBAT:

−Identify the degree and leading coefficient of polynomials

−Determine the end behavior of a polynomial from it’s equation

−Find the real zeros of a polynomial from a graph, a table, and by factoring

−Sketch a graph of a polynomial from a function

Polynomial Functions

General Form:

is an integer greater than 0

the largest exponent, is called the degree of the polynomial

is called the leading coefficient

Graphs of polynomials are:

Continuous: no breaks, holes, or gaps (you don’t pick up your pencil when drawing)

Smooth with rounded turns: no points or sharp turns

The Leading Coefficient Test: If n is odd

If a is positive If a is negative

The Leading Coefficient Test:

If n is evenIf a is positive If a is negative

Apply the Leading Coefficient Test to determine the end behavior

1)

2)

3)

4)

Zeros of a Polynomial Function

For a polynomial function of degree , the following statements are true.

1) The function has at most real zeros

2) The graph of has at most relative maximums or minimums

Real Zeros of Polynomial Functions

If is a polynomial function and is a real number, the following statements are equivalent:

1) is a zero of the function

2) is a solution of the equation

3) is a factor of the polynomial

4) is an x-intercept of the graph of

Find the zeros of each function

1)

2)

3)

4)

Repeated Zeros

For a polynomial function, a factor of , yields a repeated zero of multiplicity

If is odd, the graph CROSSES the x-axis at

If is even, the graph TOUCHES the x-axis (but does NOT cross the x-axis) at

Writing a Polynomial Function from the zeros

Given the zeros 3, 3, -4, write a polynomial function.

Intermediate Value Theorem

Let and be real numbers such that . If f is a polynomial function such that then in the interval takes on every value between and

Approximating the Zeros

Using the table feature on your calculator, find three intervals of length 1 in which the polynomial is guaranteed to have a zero

Sketching the graph of a polynomial

1) Apply the Leading Coefficient Test to determine end behavior

2) Find the zeros

3) Find additional points

4) Draw the graph

Sketch the graph

1. 2.