Ch. 3.2 Polynomial Functions and their Graphs

The script begins with an introduction to polynomial functions and their graphs, setting the stage for a detailed exploration of how to manually graph these functions. The instructor emphasizes the relevance of this skill in a calculus course, where more advanced techniques for graph analysis will be introduced. The session also mentions the introduction of calculus notation to facilitate the discussion on functions and other mathematical concepts throughout the course. A polynomial function is defined as an expression of the form p(x) = a_nx^n + ... + a_1x + a_0, highlighting the role of coefficients and the significance of the degree of the polynomial, which dictates the general shape of the graph.

This paragraph delves into the characteristics of polynomial graphs, focusing on the degree of the polynomial and its impact on the graph's shape. It explains how the degree correlates with the maximum number of x-intercepts (zeros) a polynomial can have. The instructor also discusses the distinction between odd and even degree polynomials, noting that odd-degree polynomials tend to have a linear-like shape with 'squiggles', while even-degree polynomials resemble a 'U' shape. The concept of end behavior is introduced, describing how the graph of a polynomial behaves as x approaches positive or negative infinity, which is determined by the sign of the leading coefficient and the degree of the polynomial.

The script continues with a deeper look at the end behavior of polynomial functions, using mathematical notation to describe the limits of these functions as x approaches positive or negative infinity. It reiterates the patterns observed in odd and even degree polynomials, emphasizing how the sign of the leading coefficient influences whether the graph opens upwards or downwards. The explanation includes a brief discussion on the implications of the leading coefficient being positive or negative on the direction in which the graph's ends point.

The importance of zeros in polynomial functions is highlighted in this section, explaining how they help shape the interior of the graph. The concept of multiplicity is introduced, which refers to the number of times a zero occurs in a polynomial. The instructor clarifies that an odd multiplicity results in the graph crossing the x-axis, while an even multiplicity causes the graph to 'bounce off' the axis. The summary also touches on the implications of the leading coefficient's sign on the end behavior of the graph.

The script provides a step-by-step approach to graphing polynomials by hand, starting with understanding the degree of the polynomial and its general shape. It discusses the significance of the leading coefficient in determining the end behavior and the role of zeros in plotting the graph's interior. The concept of local extrema is introduced, explaining that a polynomial of degree n can have at most n-1 local maxima or minima. The instructor uses a visual graph to illustrate these points and emphasizes the importance of graphing polynomials by hand to develop a deeper understanding of their behavior.

This paragraph outlines the detailed process of graphing a specific polynomial function by hand, emphasizing the importance of understanding the degree, end behavior, y-intercept, and zeros of the function. The instructor demonstrates how to calculate the y-intercept and how to determine where the graph crosses or bounces off the x-axis based on the multiplicity of the zeros. The summary includes a rough sketch of the graph based on the information gathered and suggests evaluating the polynomial at interim points for a more accurate representation.

The final paragraph compares the rough graph sketched by the instructor with the actual graph of the polynomial function, noting the close resemblance and the effectiveness of the manual graphing method. It suggests plugging in values at interim points between known zeros to refine the graph's accuracy, providing examples of such points. The instructor emphasizes the educational value of graphing polynomials by hand, even with the availability of calculators, for a better understanding of polynomial behavior.