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

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5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior.

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Page 1: 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.

Page 2: 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

Page 3: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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

Page 4: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 5: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 6: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 7: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 8: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

Polynomial Graphs

Page 9: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 10: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 11: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

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.

Page 12: 5-1 Polynomial Functions Classify polynomials by describing the degree and end behavior

Assignment• Odds p.285 #9-39