![]() Using the fact that these graphs are continuous for points where the function is defined, students find equations for the slope of the tangent to a rational function graph at a point in much the same way as they did for polynomial functions. As with polynomial functions they look at many graphs and develop a general description of the properties of rational function graphs, including the types of discontinuity that can occur and the behavior of these graphs for extremely large and small values of x. ![]() Students analyze the graphs of rational functions. Along the way, they develop an informal understanding of what the slope of this tangent means about the polynomial functions. They also use iterated long division, the method of undetermined coefficients, and technology to write the Taylor expansion for various polynomial functions about a point, and use this expansion to find the equation for the tangent to a polynomial graph at a point. They calculate the slope of the line tangent to the graph of polynomial functions at a point by finding the limit of the secant slope as one of the two points that determines the secant line approaches the other. Through key theorems, including the Intermediate Value Theorem, students draw conclusions about the number of roots possible for a polynomial function of given degree. Continuity is defined, and students develop both a formal and an informal understanding of what it means. Students look at the graphs of many different polynomial functions and develop a general description of the properties of these functions, especially for polynomials of degree three and four, including their behavior for extremely large and small values of x. ![]() Chapter 3: Analysis of Functions 3A: Analysis of Polynomial Functions ![]() This study is especially suited for showing the connections between algebra, geometry, and analysis: using trigonometric functions to represent complex numbers allows one to use analytic methods to solve algebraic problems (and vice versa), connecting the roots of xn – 1 = 0 and the regular n-gon allows students to use the power of algebra to solve geometric problems and to use geometric methods to get algebraic insights, calculating with "cyclotmic integers (complex numbers that are linear combinations of nth roots of unity with integer coefficients) shows once again how one can model algebraic systems with polynomials and computer algebra systems. In this investigation, students study the roots of unity. Students learn that by using complex arithmetic, they can easily prove trigonometric facts that can prove quite difficult to show strictly using algebra and geometry of the real plane. In this investigation, students explore ways to build and prove trigonometric formulas and identities. The exercises lead up to the Multiplication Law for complex numbers, which highlights the connection between complex numbers and transformational geometry: a vector can be dilated and rotated by performing one multiplication. In this investigation, students explore polar form as a way to represent complex numbers. Chapter 2: Complex Numbers and Trigonometry 2A: Graphing Complex Numbers The lessons de-emphasize the memorization of formulas plucked from coefficients - instead, amplitude and phase shift are motivated with concrete examples. This investigation emphasizes the idea that trigonometric functions are "functions as usual" - once students are familiar with the graphs of the functions defined by f(x)=cos x and f(x)= sin x, the sinusoidal functions emerge from the same transformations that students used beginning in Algebra 1. 1C: Trigonometric Functions and their Graphs They also review the definitions of inverse function and one-to-one, and learn how to restrict the domain of cosine, sine, and tangent in order to define their inverses. They review the graphs of these functions, learn about how each function can be demonstrated on the unit circle, and work with some basic identities. In Investigation 1B, students round out the list of trigonometric functions, learning tangent and the three reciprocal functions, secant, cosecant, and cotangent. Finally, they begin to solve equations that involve cosine and sine. They then review the graphs of the cosine and sine functions, and develop an understanding of a periodic function. They do work on the customary problems of converting between degrees and radians, but the goal is to enforce the concept that a radian is a distance measurement, not another way to measure angles. 1 radian, then, is an arc of the unit circle with length 1 unit. The text is careful not to equate degree and radian measurement (such as one would equate inch and centimeter measurement): radian measure is the length of an arc of the unit circle. This investigation introduces radian measure. Chapter 1: Analyzing Trigonometric Functions 1A: The Cosine and Sine Functions
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