Studied by 2 peopleLeading 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
Note
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