Superhero transformations

Superhero transformations was an assignment that was supposed to help us remember and learn how to use the different functions of a line. There were seven different functions or superheroes that we used were linear, quadratic, power, ploynomial, rational, exponential, and logarithmic.a linear equation is a straight line and is identifies as f(x)=ax+b. A quadratic equation can be identified as y=x^2 and the graph looks like a U, otherwise known as a parabola.  Power function is exactly that, a function with a power and it's equation looks like f(x)=ax^b. An example of a polynomial function is y=x^2+x+b? A rational equation looks like y=2x+5/x-1. An exponential graph looks like a steep curve and is represented by a fixed base and a variable as an exponent unlike a power function. A logarithicmic function is y=log_2(x).

Verifying Trig Identities

Verifying, or proving, trig functions is only a matter Of plugging and chugging. There are some suggestions that you should follow though when verifying a trig function. 

1. Simplify the more complicated side: the equation will look something like sinx=sinxcosx. So the right side would be better to simplify because it is more complicated.

2. Find the common denominators: this is a relative rule that applies to almost all maths and should be common sense by now.

3. Change all trig functions in terms of sin and cos: sin and cos are far easier to work with than tan which is sin/cos anyways.

4. Use an identity: an identity is exactly that- an identity! It's very similar ton the transition property used in geometry where if A = B and  B= C, then A = C. 

Tangent

Tangent is the trig function is the relation of the opposite side over the adjacent side. This is seen in the TOA part of SOH CAH TOA. Unlike the sin and cos functions of trig which are fairly similar in image, the graph of a tan line is completely different. The period of tangent is pi instead do two pi because the middle of the function goes up and through the origin. The graph also has vertical asymptotes so that each line is distinguishable. 

The graph goes infinitely up and down and cross the x-axis every pi instead of two pi as clearly shown in the picture above. 

Sine and cosine

A trig function is the function of an angle and relate to the angles of a triangle to the length of its sides. Sine is the trigonometric function that is equal to the ratio of the side opposite of a given angle to the hypotenuse in a right triangle. Cosine is the trigonometric function that is equal to the ratio of the adjacent side to the hypotenuse of a given angle in a right triangle. The ratios can be easily remembered through the acronym SOH-CAH -TOA. The first letters of each group stands for the trig function (sin, cosine, or tangent). The two letters following represent the ratios. So Sin(Opposite/Hypotenuse)=SOH and so on. 

The graph of sine begins at the origin while the graph for cosine begins at 1 or your amplitude. 

Chapter 3 Summary

In chapter 3, we basically went over almost all of the functions. We learned his to dividen ploynomial functions and then finding the zeros and factors of polynomial functions. An important fact to note when looking for question regarding then factors/zeros of a polynomial function is that a "root" is the same and interchangeable with "zero". We the went over the real zeros of a polynomial function which is basically funding deriving the zeros from the factorization. We learned to approximate zeros and the moved on to rational functions which is any function that can be define by a rational fraction.

This is an example of a polynomial function. 

Rational functions

A rational function is any function which can be defined by a rational fraction where both the numerator and denominator are polynomials. To graph a rational function, you aphave to first find the asymptotes and the intercepts. Then from that, you have to plot the plots then graph. The vertical asymptote can be found by setting the denominator equal to zero and the solving for x. Once the asymptotes are drawn on the graph, it means that the function cannot touch that line so the graph would maneuver around it. The horizontal asymptote can be found by comparing the exponents of the variables. So the largest variable in the numerator is compared to that if the denominator by following a set of rules. If n<m, n being the numerator, then the x-axis is the horizontal aymptote. If it is the opposite, the there is no horizontal asymptote and instead you would find a slant asymptote. If n=m, then the horizontal asymptote is y=a/b. 

Zeros of a Function

The zero of a fucntion is an x-value that makes a function into zero. It can also be called a root of a function of f or an x-intercept. The zero of a function can be found  by solving the equation for x. Whatever the x value turns out to be will be the zero of the graph because at that point, if plugged into the function, it will cause the entire graph to be zero. Roots, zeros, and x-intercepts can all be real or complex. An example would be x^2=4. If one was to solve for x, then the square root of 4 would be 2, and that 2 would be the zero of the function.