Chapter 4 paper
In this chapter one of the first things we learned was that you can make complex equations look simpler by making substitutions. Take the equation: 2tan2-3tan-2=0
To make this look simpler you can substitute u for tan to get: 2u2-3u-2=0
From there you can us the quadratic formula to get:
u= -0.5, 2
With this you can now say tan=2 and tan=-0.5 which makes solving for much easier. In this case substitutions made solving much easier and less chaotic looking.
The next thing we learned was to rewrite equations in a way that suits your purpose. One key skill for rewriting equations is to get rid of negative exponents. The equation (x-1+y-2 )/(x+y) can be simplified to get rid of the negative exponents. X-1 =1/x therefore: (x-1+y-2 )/(x+y)=(1/x)+(1/y2/(x+y)
From there you can simplify it further by multiplying the top and bottom by a common factor, in this case it is: x+y2 So to finish simplifying the equation you take:
And multiply it by (x+y2)/(x+y2) to get:
Other common rewrite is to rationalize the denominator or numerator. This is very helpful for solving equations with a square root on the bottom. Example: (3)+2/(3)+1
To rationalize this you multiply the numerator or denominator (which ever one you want to rationalize) by its difference of squares. For this:
To rationalize the denominator you multiply it by ((3)-1)/ ((3)-1) Giving you: 1+3/2.
Another thing we learned was how to tell when a graph is concave up or down and decreasing or increasing. A graph is concave up when the x values are above the x axis and concave down when they are below. A graph is Increasing when you increase the y value and the x value increases and its decreasing when the x value decrease.