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.
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