Tangent Graph

The tangent function has a completely different shape, it goes between negative and positive infinity, crossing through 0. At pi/2 radians, or 90 degrees the function is officially undefined, because it could be positive infinity or negative infinity. For a tangent graph one cycle occurs between -pi/2 and pi/2. There are vertical asymptotes at each end of the cycle. The asymptote that occurs at pi/2 repeats every pi units. The period is pi and there is no amplitude because the graph continues on forever in vertical directions. 

Tangent graph:

Sine and Cosine Functions

For each real number t there corresponds an angle of t radians in standard position. We denote the point of interesection of the terminal side of the angle t with the unit circle by P(t). The x and y coordinates of this point give us the values of the six basic trigonometric functions. The y coordinate of P(t) is called the sine of t, while the x-coordinate of P(t) is called the cosine. The sine definition basically says that, on a right triangle, the following measurements are related the measurement of one of the non-right angles, the length of the side opposite to that angle and the length of the triangle's hypotenuse. Alternately, the cosine definition basically says that, on a right triangle, the following measurements are related: the measurement of one of the non-right angles, the length of the side adjacent to that angle and the length of the triangle's hypotenuse. Therefore cosine equals adjacent over hypotenuse and sine equals opposite over hypotenuse. 

Chapter 3 Summary

For Chapter three there are many different methods such as Approximate- midpoint, rational zero test, remainder theorem, long division, synthetic division, conjugate theorem, and number of zeroes which is degrees. When given a real zero in a problem there are three different steps to do: 1. Division (synthetic or long division), 2. Continue dividing, 3. Quadratic- either quadratic formula or factoring. When given imaginary zeroes in a problem there are also three different steps: 1. Write conjugate pairs as factors, 2. Multiply, and 3. divide. If your given nothing in a problem then you must use the rational zero test, test all possibilities and find one that works. When doing synthetic division remember to always switch the sign of the number you are using or else your answer will be wrong! Also, for the remainder theorem, the number you are given, you can substitute it into x. For the rational zero test, once you find a number that works, you must also find other zeroes that work too! 

Picture of me doing a problem involving long division: 

Rational Functions

Rational Functions are any functions which can be defined by a rational fraction for example an algebraic fraction such that both the numerator and the denominator are polynomials. A function  is called a rational function if and only if it can be written in the form

. Where  and  are polynomials in X and Q is not the zero polynomial. The domain of f is the set of all points X for which the denominator  is not zero. However, if P and Q have a non constant polynomial greatest common divisor R, then setting  and  produces a rational function . The degree of a rational function is the maximum of the degrees of its constituent polynomials P and Q. Rational functions with degree 1 are called Mobius transformations and form the automorphisms group of the Riemann Sphere. Rational functions are representative examples of meromorphic functions. 

Graph of a rational function: