The Tangent Graph

Tangent is another function that is similar to the ways that sine and cosine work but the graph is much different. Unlike sine and cosine, tangent requires asymptotes which show where the graph cannot cross. Also each period of a tangent graph is not connected but are there own separate graphs that go up and down infinity. The tangent equation is y=Atan(Bx+C) + D. To find asymptotes of a tangent graph use the equation pi/2+npi. The equation to find the period is pi/b and to find the x intercepts is Bx+C=npi and A is the amplitude. Here is a picture of a tangent graph.  

Sine and Cosine Functions

Sine and cosine are functions that are used to find angles and sides of triangles. On a graph cos=x and sin=y. To use sin and cosine you use the acronym SOH CAH: soh means sin equals the Opposite side from the angle over the hypotenuse of the triangle. CAH means cosine equals the adjacent side of the triangle over the hypotenuse of the triangle.  Secant and cosecant are two more functions that relate to cosine and sine.  Secant equals 1/cos and cosecant equals 1/sin. Basically they are the reciprocal of cosine and sine. Here is a picture showing how to use sine and cosine in a triangle.  

Summary of Chapter 3

The main focus of chapter 3 was to find the zeroes of a function and there are many steps and different ways to do that. First we learned about polynomial functions and looked at end behavior and how to tell if a function is even or odd. We also learned about multiplicity which tells how many zeros a function has. We also learned how to do division of a polynomial function using long division and synthetic division which can be used to find the zeroes of a function. Another way to find zeroes is using p/s where p equals the factors of the constant over the factors of the leading coefficient and then you find all possible zeros and test those zeros in the equation. We also learned about approximating zeros which is basically just dividing the interval in half until you get a zero. The last thing we learned is rational functions and we learned how to find vertical, horizontal, and slant asymptotes and the holes of an equation. Here's a picture showing how to find all these asymptotes and holes. 

Rational Functions

A rational function is a polynomial divided by a polynomial, basically a fraction with polynomials in the numerator and denominator.  Examples are x^2+3x-2/3x^2+8x-7 and 2/x-7 or even 3x^2-5. However rational functions are not allowed to have square roots or fractions in them like 3-sqrt 4/6+x. Rational functions can generally be expressed as f(x)= p(x)/q(x) as long as q(x) does not equal zero.  To find the roots of a rational function you have to find where the graph crosses the x axis or the x intercepts of the equation. To find the vertical asymptotes you just set the denominator equal to zero and solve for x. For horizontal asymptotes if the degree of the numerator equals the denominator then you divide the leading degree of the numerator and denominator, if the degree of the numerator is less than the denominator than y=0 and if the degree of the numerator is greater than the degree of the denominator then the horizontal asymptotes do not exist. Here is an example of a rational function graphed.

Finding Zeros of a Function

The zeroes of functions is where the graph crosses x. To find zeroes of a function you set the equation equal to zero. Another way to do/say this is to set y equal to zero. Once you do this simply solve the equation for x. Some times you will get more than one answer and that's okay. When you get the same number multiple times that number has multiplicity. For example if you get the number two three times you would say 2 has a multiplicity of three and if 5 appeared seven times you would say five has a multiplicity of seven. Here is an example of finding zeroes. 

Piece Wise Functions

A piece wise function is when a function has two different formulas that are both defined on the domain of f. Even though they are two different equations they are looked at as one function. They are usually used to represent the negative and the positive numbers. For example one equation usually has the restriction of x<0 and the other equation is x>0. To graph a piecewise defined function you graph each equation as you would normally and then it is either considered continuous or discontinuous. A continuous graph is when you don't have to lift your pencil when tracing the graph. A discontinuous graph is disconnected in places. Here is an example of a piecewise function problem.

Superhero Transformations

Today I worked on superhero transformations. There are six superheroes, lady straightedge, pawabawa, robo-grow, captain abs, radical girl, and bipolar tommy. Lady straightedge represents a straight line, her equation is f(x) = x and she has nuclear fusion blasts from her eyes. Pawabawa is a parabola his equation is f(x) = x^2 and he has parabolic kinetic rays. Robo-grow is from the exponential family, her equation f(x) = 2^x and she has a ginormous robot suit. Captain abs represents absolute value his equation is f(x) = |x| And he has finger beams. Radical girl represents a square root her equation is f(x) = sqrt x and he has ninja skills with a sword. Lastly bipolar tommy represents Cubing and his equation is f(x) = x^3 and he has a curvy laser gun. All these superheroes and their powers helped to complete the missions.