Warm-Up Feb. 10th State the end behavior (no calculator) and the domain and range for each polynomial.

1. f(x) = x2 + 4x – 11

2. g(x) = x3 + 4

3. h(x) = 7 – 3x5 + 6x

GRAPHING & WRITING

POLYNOMIAL FUNCTIONS

Vocab & Background The maximum number of Extrema (max and

mins) of a graph can be found from the degree.If the degree of the polynomial is n, then the

number of extrema is at most n – 1. (there may be less)

The degree will also tell you the maximum number of possible x-intercepts. (there may be less)

Zeros (aslo known as roots or x-intercepts) are where the graph crosses the x-axis.In some cases there may be a double zero, meaning

the graph bounces at that value. There may also be imaginary roots which don’t

show up on the graph but still count toward the maximum number of zeros.

Y-intercept is where the graph crosses the y-axis (where x = 0)

P(x) = 5x4 + 3x3 – 4x5 + x + 6x2 – 2

Degree:

Leading Coefficient:

Max Number of Extrema:

Max Number of Zeros/X-intercepts:

Y-intercept:

Increasing/Decreasing Reading the graph from left to right (very

important!)…The graph is increasing where the y-values are

increasingThe graph is decreasing where the y-values are

decreasing It changes at its extrema. We use interval notation to write where the

graph is increasing or decreasing.

Parts of a Polynomial Graph Extrema ~ Max/Min

absolute vs. local (relative)

Y-intercept:

X-intercepts/Zeros:

Increasing:

Decreasing:

Parts of a Polynomial Graph Extrema ~ Max/Min

absolute vs. local (relative)

Y-intercept:

X-intercepts/Zeros:

Increasing:

Decreasing:

Examples For the function below find the following: y-intercept,

end behavior, domain, zeros, range, extrema and intervals of increasing/decreasing.

*Hint: there may be multiplicity of zeros. Also if you must round, round to the nearest hundredth.

10132 23 xxxy

234 82 xxxy

22 26 xxxy

You Try:24410 234 xxxxy

xxxy 54 23

Writing Polynomials1. Zeros at -2, 3 and 5

2. Zeros at 4, -6, and 0.

3. Zeros at 1 and ¾.

Double Roots A zero with a

multiplicity of 1 will cross the x-axis.

A zero with a multiplicity of 2 will “bounce/touch” the x-axis.

From a Graph

Special Roots An irrational root of a polynomial contains a radical.

For example: They always travel in pairs with their conjugate.

An imaginary root contains an imaginary number, i.For example:They also travel in pairs with their conjugate.

7 52

ii 4 23

Examples with Special Roots Write a polynomial for the given roots

3 and 5 ,2 .2

11and4 .1

i