math final notes (sem 2)

module 7: rational function

if the problem is simplifying and you need to add, you can’t get rid of a common factor because it is addition
an expression is undefined (excluded value) when the denominator is equal to 0
graphing rational functions
- vertical asymptote: if f(x)=a(x)/b(x) and there are no common factors other than 1 for a(x) and b(x), solve for x in the denominator to get V.A.
- horizontal asymptote:
  - if degree of a(x) is greater than degree of b(x) —> no H.A.
  - if degree of a(x) is less than degree of b(x) —> H.A. is y=0
  - if degree is equal, H.A. is y= a(x) leading coefficient/b(x) leading coefficient
- oblique asymptote: if degree of a(x)-degree of b(x)=1, there is an oblique asymptote
  - equation of asymptote is y=a(x)/b(x) with no remainder (use long division)
- holes/point discontinuity: if f(x)=a(x)/b(x), b(x)≠0, and x-c is a factor for both, there is a hole at x-c
- multiplicity: number of times a factor/number shows up
- even/odd function:
- number line things (?):
when solving rational equations, find the common denominator and use cross multiplication with answer to solve for x
- first step however should be to find the excluded values

module 4: inverse & radical functions

when doing (f/g)(x), you can just leave the answer in the (f/g)(x) form as long as you write the excluded values
- or you can use synthetic division or long division to solve
when checking inverses, you can use composition of functions to find
- composition of functions: plug one function into another and if it =x, it is an inverse
changing exponent to radical form: the numerator is the number inside the rad sign and denominator is the power of the radical (?)
graphing radical functions
- when the function is like square root, the end point is the numbers
  - ex. g(x)=-(sqrt x+6)-2, then the end point is going to be at -6,-2 because it is translated down 2 and moved left 6 units
  - find the next point by plugging in a point
- when the function is like cube root, the middle point is the numbers
  - ex. g(x)=(cbrt x)-4, then the middle point is down 4 and x-value is still at 0
  - find the next point by plugging in a point
  - the graph is shaped like a S

module 5: exponential functions

formula for continuous compound interest: A=Pe^rt
formula for continuous compound interest: A=P(1+r/n)^nt
if something is increase by a certain percent each week or something, you would multiply the starting factor and (1+%)^x
- refer to question 14
e is around 2.718

module 6: logarithmic functions

rewriting exponent in logarithmic form
- snail method
when solving for x given equation but the number in the equal sign doesn’t have a log, you have to make the base of the log to the power of that number after that equal sign
- ex.
ln is the same as log e
log 10 is the same as e
log functions are inverses of exponential functions
log functions do not have horizontal asymptotes
- they only have vertical and you find that by looking at the function and if there is a + or - next to the x, that is the vertical asymptote)
how to graph
- find two points on the parent function
- do the reflection/dilation
  - if - is outside the log then you change the y-value
- then do the translations
- the order is similar to PEMDAS
when solving for x given equation, usually - answers are extraneous because you can’t have a negative argument (the thing inside the () after log base __

module 13: trigonometric identities

module 8: inferential statistics

other notes