Student+Notes+10+Polynomials+End+Behavior+and+Continuity

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California State University, Long Beach **We aren't endorsed by this school
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MATH 113
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Mathematics
Date
Oct 27, 2023
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5
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Math 113 Lecture Notes 10: Polynomials 3.2 1 3.2 Polynomials and their Graphs Students identify graphs of monomials x n . Students identify features that cannot appear on the graph of a polynomial. Determine the end behavior of a polynomial from its formula and from its graph. Use correct notation to describe the end behavior of polynomials. Monomials Graphs of monomials (Figure 1 Section 2.2) - Label each graph with its formula. Using techniques from Section 2.5, we can transform these monomials (Figure 2 Section 3.2). We add and subtract these transformed monomial functions to get polynomial functions. Polynomials The graphs of polynomials look like this, for example: These graphs are continuous and smooth . Graphs of continuous functions have... Graphs of smooth functions have... These graphs do not represent polynomials: What is the difference between a cusp and a corner?
Math 113 Lecture Notes 10: Polynomials 3.2 2 Vocabulary A polynomial is a function of the form , where n is non- negative and . The numbers a 0 , a 1 , ... , a n are called the ____________________________ of the polynomial. The number a 0 is called the _____________________________ or ______________________. The number a n , the coefficient of the highest power, is called the _________________________, and the term a n x n is called the __________________________. Example This is a polynomial of degree _________. Coefficients a 0 = a 1 = a 2 = a 3 = a 4 = a 5 = a 6 = a 7 = constant coefficient leading coefficient leading term End behavior When we look at the end behavior of a polynomial, we are describing the output of the polynomial as the input x gets large in the positive or negative direction. Description: x gets large in the positive direction x gets large in the negative direction Numerical example: An example of the inputs x "getting large in the positive direction": x starts at ____ then increases to ___, to ____, to _____ to _____ and on up toward_______. An example of the inputs x "getting large in the negative direction": x starts at ____ then decreases to ___, to ____, to ____, to _____ to _____ and on down toward_______. Notation: Label each of the following graphs with its end behavior P ( x ) = a n x n + a n 1 x n 1 + ... + a 1 x + a 0 a n 0 f ( x ) = 2 3 x + 4 x 2 + 10 x 5
Math 113 Lecture Notes 10: Polynomials 3.2 3 Numerical Example Consider the polynomial . x 1 -1 3 4 6 10 -100,000 3,000 40 -96,960 100 -10,000,000,000 3,000,000 400 -9,996,999,600 1,000 -1E+15 = _________ 3,000,000,000 4,000 10,000 -1E+20 =__________ 40,000 as x ® - x 5 ® 3 x 3 ® 4 x ® f ( x ) ® In this example, which term dominates? _____________. What is the end behavior displayed here? In a nutshell The end behavior of the polynomial is determined by the degree n and the sign of the leading coefficient a n . P ( x ) has odd degree ( n is odd) Leading coefficient is positive ( a n > 0) Leading coefficient is negative ( a n < 0) Example formula: Example formula: End behavior: as as End behavior: as as f ( x ) = x 5 + 3 x 3 + 4 x x 5 3 x 3 4 x f ( x ) = x 5 + 3 x 3 -9.99997 × 10 15 3 × 10 12 1 × 10 20 P ( x ) = a n x n + a n 1 x n 1 + ... + a 1 x + a 0 y ________ x + y ________ x → − y ________ x + y ________ x → −
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