5-1 polynomial functions classify polynomials by describing the degree and end behavior

5-1 Polynomial Functions

Classify polynomials by describing the degree and end behavior.

Monomials• A real number, variable, or product of

numbers and variables.• A single term

o No addition or subtraction

• Examples:o 18o Zo -4x2

o 2.5xy

Degree of a monomial

• Sum of the exponents of its variableso Degree of a constant with no variables is 0

• Ex:• 5x

o Degree 1

• 6x3y2

o Degree 5

• 4o Degree 0

Polynomials• Monomial or sum of monomials• Terms separated by addition or subtraction• Ex: • Standard form of a polynomial is when

the polynomial is arranged so that the degrees of each monomial decrease from left to right.

Degree of a Polynomial

• Degree of the monomial with the highest degree.

• Ex: what is the degree of the polynomial? o 4

• Polynomials are named according to number of terms.

Practice• Write the polynomial in standard form.

• Place in order: • Combine like terms: • What is the degree?

o 3

• How many terms?o 3

• This is a cubic trinomial.

Polynomial Graphs• The degree affects the shape of the graph.• It also determines the max number of turning

points.o Places where the graph changes direction.o There are a max of 1 less turning points than

the degree. (quadratics have 1, cubics have 2, quartics have 3, etc.)

• End behavior tells the direction of the graph to the far left and far right.

• A function is increasing when y-values increase as x-values increase.

• A function is decreasing when y-values decrease as x-values increase.

Polynomial Graphs

Practice• Describe the end behavior

• End behavior is down and up• Odd exponents with positive a-value go down and

upo If it has a negative a-value then it switches to up and down.

• End behavior is down and down• Even exponents with positive a-value go up and

upo If it has a negative a-value then it switches to down and down.

Graphing Cubic Functions

• Use a table with some negative x-values, 0, and positive x-values

• The graph has no turning points (it is always increasing from -∞ to ∞) and end behavior is down and up.

Using Differences to Determine Degree

• Find the degree of the polynomial function that generates the data shown.

• List y-values vertically.• Find the difference between each y-value• Do it until the differences are all the same.• Values match at 3rd

difference.• Degree is 3.

Assignment• Odds p.285 #9-39