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Calculus Prerequisites, Connection: Odd functions are shown as a sine…
Calculus Prerequisites
Functions
Even functions
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Depend on the power of x. Must be positive (x^2,x^4, etc.) will result in an even function because (-x)^2 will equal a positive value
Odd functions
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Depending on the power of x a function will be odd. If the power is odd (x^3, x^5, etc.) it will be an odd function. When plugging in (-x)^3 the answer would be negative which proves this
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One-to-One funtions
Horizontal line test. If a graph is a function conduct a horizontal line test this time to see if it passes one point. if not it's not a one-to-one function. If f(x1)=f(x2) it is a one-to-one function
The inverse of a one-to-one function: Change the f(x) into y and once there is a y=mx+b equation switch the x and y values to make it x=my+b. Isolate y and that is the inverse
Ways to represent functions: 1. Function Machine
- a "mapping"
- Table of values
- A graph
- An equation
Piece-wise functions: 
When graphing a piece-wise function look at each line individually. The second part of each line is the domain which will base the table of values and the first part is the equation used to create the table of values. Plug numbers into the 3 equations and use that to the graph. *note: each domain is given in each line so when choosing points for the table follow the requirements.
Open interval: ( ). Closed Interval: [ ] Open/Closed interval: [ ), ( ]
When looking at the piece-wise functions, use the same concept used for intervals. Open circle or closed circle
Composition of functions
(gof)(x) = g(f(x)), (fog)(x) = f(g(x))
For g(f(x)), plug the x value into the f(x) value which should be given and once the answer is found, put that into the g(x) value. Do the opposite for f(g(x)) plug the x value into g(x) and then plug into f(x) to get the answer
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Lines
Linear Functions:
Δx=x1-x2, Δy=y1-y2
Slope: Δy/Δx --> y2-y1/x2-x1
To find the increments (or change in the x/y value) use the formulas shown. These formulas can also be written as Δx=x2-x1, Δy=y2-y1
Parallel and Perpendicular lines: Parallel lines have the same slope (m1=m2), and perpendicular lines have inverse reciprocal (m1 x m2 = -1) slopes
Equation with points given: Use the given values and use the slope formula to find the 'm' value and plug into either the point slope equation or the slope intercept form. If general form is needed, rearrange the formula to get the x and y value on the same side
Equations for parallel and perpendicular: Parallel lines will have the same 'm' value (m1=m2). For the next value plug in all given values into the point-slope equation. Perpendicular lines need to multiple and equal -1 (m1 x m2 = -1) and once the m value is found (reciprocal of given) find the rest by using point-slope as well
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Can perform line tests for parallel and perpendicular lines to see if the graph drawn is parallel or perpendicular
Exponential Functions
To graph functions with variables in power for example 2^(x+3) plot the graph regularly (2^x) by plugging in values for x and use translation to move. Since it is x+3 it will be moved 3 units down so shift the graph down
Compound interest
A= Ao(1+i/n)^nxt
A= future amount
Ao= initial amount
i= interest rate
n= how is it compounded?
t= number of years
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Parametric Equations
Relations: If x and y are functions and there is a third variable (t) called parameter, use parametric mode to get the graph
If 't' value, choose numbers depending on the domain as 't' and plug those into the x and y equations
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Trigonometric Functions
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Trigonometric Ratios:
3 basics: sin0=y/r, cos0=x/r, tan=y/x
3 reciprocal ratios: csc0=r/y, sec0=r/x, cot0=x/y
When finding all trig values use the given value and plot it on a graph. Use the ASTC table shown above to know which quadrant each of the basic ratios are positive in. Find the unknown value if needed and once the x,y,r values are found plug them into each trigonometric ratio
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Logarithm functions
logx^n = nlogx <-- Power Rule
log(mxn) = logm+ logn <-- Multiplication Rule
log(m/n) = logm-logn <-- Division Rule
B^k=A <=> logbA=K <-- Definition of log
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Horizontal line test is the same as the vertical line test used to determine a function, just the other way (horizontal instead of vertical)
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*note if the equation is not an odd function or even, label it as "neither"
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