Sine and Cosine #9

Sine of theta is equal to the y value of a triangle. Cosine of theta is equal to the x value of a triangle. A way to find sine and cosine is to remember SOH CAH. The SOH represents sine = opposite/the adjacent side of the theta. The CAH represents cosine = adjacent/the hypotenuse of the triangle. Sine is always positive in the first and second quadrants. Cosine is always positive in the first and fourth quadrants. 

Chapter 3 Summary #8

Chapter 3 covered polynomial functions, multiplicity, synthetic and long division, the rational zero test, approximating zeros, and rational functions. The equation for a polynomial function is: AnX^n + An-1X^n-1 + An-2 + A1X^1 + Ao. A is the coefficient and ^n is the degree. An example of multiplicity is x^2. You would say is as x with a multiplicity of 2. For synthetic division, the first step is to do complete factorization, then list all the zeros. a+bi is a complex number, a-bi is a conjugate, fl them together. The rational zero test us all the factors of the constant (P) / factors of the leading coefficient (S). The first step for approximating zeros is to divide the interval [a,b] in half by finding its midpoint, m=a+b/2. Then, if f(a) and f(m) have opposite signs, then f has a zero in the interval [a,m]. If f(m) and f(b) have opposite signs, then f has a zero in the interval [m,b]. If f(m) = 0, then m is a zero of f. Finally, compute f(m). 

Piecewise Functions #5

A continuous piecewise function has no holes, no gaps, and is drawn without lifting your pencil. A discontinuous piecewise function is the exact opposite and has an end. The absolute value function has two pieces: below zero (-x) and from 0 onwards (x). For this type of piecewise function, once you graph the first side of it, you know the absolute value side, because it is a reflection from the first half of the graph. Given the title "piecewise," a graph can be in multiple pieces and not be connected at all! The greatest integer function (aka the floor function), has an infinite number of pieces and looks like steps or stairs when graphed. The pattern is continuous. 

Superheroes #4

I didn't really understand how to complete all of the missions, but I was able to do two of them. I think Lasy Straightedge was th best superhero. She moved smoothly along the X and Y graph and had great aim! Pawabawa was pretty cool and he also had great aim. The trickiest heros were the last four, because they were not the typical line or palabara. They made me frustrated and confused. 

Rational Functions #7

You can use the Rarional zero test in order to find zeros. First, take the final number without a variable and write out all the factors of it. This will be your numerator. Then, take all the factors of your leading coefficient and that will be your denominator. Once you have solved for all the possible zeros, you must plug each one back into the original equation. Yes, this will take a lot of time and you will want to scream as I did. When you find a zero that makes your equation zero, then you know you have found a rational zero! 

Zeros of a Function #6

For the Rational Zero Test, we first find all the possible zeros using P/S then test the zeros using the remainder theorem. A hint would be to graph to see all the possible points. The Rational Zero Tsst gives you the possibility of all the rational zeros. Once you find one, you can do synthetic division. Just like last week, we might have to use synthetic division multiple times. Since the work it really tedious, the Rational Zero Test is the best method to use. Remember both positive and negative parts of zeros!

Functions #3

Today, I learned that a function was an equation that you could put a in number called an input to end up with a solution which is the output. An example of a basic function equation would be f(x)=2x^2+1. A subsection of a function would be its domain. A domain is what you ask yourself, "what can I put in?" There are two ways to solve domains. First, if the problem has a square root you must set it to > or equal to 0, because a negative cannot be in a square root. Second, if there is denominator in the problem, set each denominator to zero, but keep in mind, the answer cannot be zero. If the answer is zero, the solution is undefined.