Section 4.1 Polynomial Functions Honors Algebra 2 Section 4.1 Polynomial Functions

Monomial-a number, a variable or a product of a number and one or more variables Polynomial-monomial or a sum of monomials Polynomial function-function in the form 𝑦= 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 + 𝑎 𝑛−2 𝑥 𝑛−2 +…+ 𝑎 1 𝑥 1 + 𝑎 0

Things about a polynomial function the exponents are whole numbers The coefficients are real numbers 𝑎 𝑛 is the leading coefficient 𝑛 is the degree of the polynomial function 𝑎 0 is the constant term

A polynomial function is in standard form when the terms are written in descending order of exponents from left to right. 𝑦=4 𝑥 4 −2 𝑥 3 +7 𝑥 2 −1𝑥+19

Decide if the following are polynomials Decide if the following are polynomials. If so, write in standard form and state its degree, type, and leading coefficient #1 𝑓 𝑥 =−4 𝑥 4 +7 𝑥 2 −2 #2 𝑔 𝑥 =.5 𝑥 2 + 3 𝑥 4 +11 #3 ℎ 𝑥 =− 𝑥 2 +7 𝑥 −1 +4x #4 𝑘 𝑥 = 𝑥 2 + 3 𝑥

For a polynomial function, one thing (x) goes in and one thing (y) comes out. f(x) is the same as y f(2) is the y value when x is 2 f(-3) is the y value when x is -3 Plug and chug to find the y value!

#1 Evaluate 𝑓 𝑥 =−2 𝑥 4 +6 𝑥 3 −3𝑥+11 when 𝑥=4 #2 For the function in #1, find 𝑓(−2)

End behavior of a function’s graph- the behavior of the graph as x approaches positive infinity or negative infinity.

When the degree of the polynomial is even, both arrows will point the same way.

When the degree of the polynomial is odd, the arrows will point in different directions.

When the leading coefficient is positive, the arrow is in quadrant I. When the leading coefficient is negative, the arrow is in quadrant IV.

Describe the end behavior of the graph of the following: #1 𝑓 𝑥 =−.3 𝑥 4 +1.7 𝑥 3 −4𝑥+6 #2 𝑓 𝑥 =2 𝑥 3 +6 𝑥 2 −3𝑥+11 #3 𝑓 𝑥 =−10 𝑥 7 +5𝑥

To graph a polynomial (3rd degree and up) #1 Make a table of values using x=0 as the middle value for x. #2 Plot those points #3 Figure out the end behavior and place arrows on your graph.

To sketch a polynomial function (3rd degree and up) #1 Analyze where the graph is increasing and decreasing (we are thinking about y values here) #2 Anytime a function changes from increasing to decreasing or vice versa, there is a turning point.

What are the intervals for when the function is increasing? Decreasing?

Real Life The estimated population P (in thousands) of people living in a city can be modeled by the polynomial function 𝑃 𝑡 =1.2 𝑡 3 −2 𝑡 2 +15.7𝑡+3.8 where t represents the year, with 𝑡=1 corresponding to 2001 Use a graphing calculator to graph the function for the interval [1,10]. Describe the behavior of the graph on this interval.

Assignment #15 Pg. 162 #1-23 odd, 27, 29, 33-36, 37-41 odd