Honors Pre-Calculus A, 4th Hour, Winter 2015: polynomial functions

Tuesday, January 13, 2015

Polynomial Functions of Higher Degree

0 Degree

1st Degree

3rd Degree

2nd Degree

4th Degree

5th Degree

Above are the graphs of varying polynomial functions. As you can see, as the degree of the function increases, the graph gets curvier. While this is not always the case, it is a general rule of thumb.

End Behavior

The ends of the graphs of the varying polynomial functions are actually pretty predictable. By looking at the leading coefficient of the term with the highest degree and the degree of the function, one can figure out how the function will look when graphed.

A positive leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach infinity. This is written as such in limit notation:

A negative leading coefficient will make the right side of the right most extreme on the graph ALWAYS approach negative infinity. This is written as such in limit notation:

The opposite end of the graph either goes the same way as the right side or does the opposite. If the degree is EVEN, the left side acts the SAME as the right. If it is ODD, the left side acts the OPPOSITE of the right side.

Zeroes

A polynomial function has as many zeroes as its degree; however, it does not necessarily have that many X-Intercepts. A polynomial may have imaginary zeroes which are not graphed on the Cartesian plane.

Sometimes roots are repeated and can make a function that has two x intercepts only have one. This is called multiplicity.

This graph should have 4 X-Intercepts, however it only has 3. At X=0 the curve runs tangent to the X-Axis

Extrema

The graph of a polynomial function may have as many relative maxima/minima as (n-1) where n is the degree of the polynomial

Review

The graphs of polynomial functions have certain rules that allow you to predict what they will look like pretty easily. Knowing the degree and the leading coefficient allows you to make a sketch of the graph that will resemble the actual graph.

Monday, January 12, 2015

POLYNOMIAL FUNCTIONS & COMPLETEING THE SQUARE

Polynomial Functions

DEFINED AS...

A polynomial function is of the form: This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

- the value of This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

must be a nonnegative integer (meaning it is a whole number and is equal to zero or is a positive integer-- no fractions or radicals!)
- all coefficients ( This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

) have to be real numbers
- the degree of the polynomial is the highest value of This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

where

- is continuous
- has a domain of all real numbers
EXAMPLE
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Completing the Square

Completing the square involves taking a polynomial and rewriting it in standard form.

Standard form of a quadratic is

For EXAMPLE... This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

To rewrite this polynomial function in standard form, we will visualize the terms of the polynomial as squares.

We need one box of This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

, 6 boxes of This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

, and 8 boxes of 1.