2.2 Polynomial Functions - HP Pre-calc

Leading coefficent:

Leading coefficent:

the coefficent of the variable with the highest exponent

Degree of a Function:

Highest exponent of any of the terms in the polynomial

Whats the name of a function with a degree of 0?

Constant

Whats the name of a function with a degree of 1?

Linear

Whats the name of a function with a degree of 2?

Quadratic

Whats the name of a function with a degree of 3?

Cubic

Whats the name of a function with a degree of 4?

Quartic

Whats the name of a function with a degree of 5?

Quintic

Whats the name of a function with a degree of 6, and so on?

Degree of 6, degree of (degree of function)

What would a function NOT look like?

Anything with sharp turns or piecewise functions

Even degree and a positive leading coefficient means:

Arrows pointing up in the same direction (U shape)

Even degree and a negative leading coefficient means:

Arrows pointing down in the same way (n shape)

Odd degree and a positive leading coefficient means:

Arrows pointing in opposite directions, with the right arrow pointing up and the left pointing down

Odd degree and a negative leading coefficient means:

Arrows pointing in opposite directions, with the right arrow pointing down and the left pointing up

Turning point:

When the graph changes from increasing to decreasing, or vice versa

If you have a polynomial of degree n, how many turning points do you have?

n-1

How many x-intercepts will you have if given a polynomial of a degree of n?

N solutions, and it can have UP TO n roots

Whats the difference between roots and solutions?

While solutions are all answers for the x-intercept (counting multiplicities and imaginary numbers), not all can be represented on a graph as roots (like imaginary numbers, as they dont touch the x-axis, so they can't be counted as x-intercepts)

Multiplicity:

Number of times a solution appears

When a multiplicity is 1 (ex: (x+1)):

Graph crosses the x-axis at that root one time

When a multiplicity is even (ex: (x+1)^2):

Graph is tangent the x-axis at that root (it touches the line, doesn't cross it, and goes back the direction it came)

When a multiplicity is odd (ex: (x+1)^3):

Graph crosses through the x-axis with a point of inflection at that root

Can a root of imaginary numbers of a polynomial function cross the x-axis?

It can NEVER cross the x-axis, therefore we dont label imaginary roots on graphs