Chapter 4, 5, 7 - AOS 3 Functions and Relations Cheat Sheet by liv.skreka

This is the standard form of a rectan­gular hyperbola:
a dilates the graph
if the graph is negative, it is reflected along the x axis.
h moves the graph left and right (along x axis)
k moves the graph up and down (along y axis)
To find asympt­otes,
y=k, horizontal asymptote
x=h vertical asymptote ( or set x-h=0, and solve for x).

First sketch the graph along asymptotes to visual whether there is a x and/or y intercept.
Then solve for x and/or y intercept and label.

First, place the denomi­nators also as the numerator.
Then, use an approp­riate coeffi­cient outside brackets to expand to the correct x value.
Once expanded, select an approp­riate number to equate to the constant.
Seperate the constant over its own fraction, and cancel out the numerator and denomi­nator.

NOTE: the order in which the transf­orm­ations are applied are very important
as above, use brackets to follow order of operat­ions.

This is the standard form of a truncus:
a dilates the graph
if the graph is negative, it is reflected along the x axis.
h moves the graph left and right (along x axis)
k moves the graph up and down (along y axis)
To find asympt­­otes,
y=k, horizontal asymptote
x=h vertical asymptote ( or set x-h=0, and solve for x).

First sketch the graph along asymptotes to visual whether there is a x and/or y intercept.
Then solve for x and/or y intercept and label.

The general form of a root x graph:
a dilates the graph
if a is negative (-a), the graph is reflected in the x axis
if x is negative (√-x), the graph is reflected in the x axis
NOTE: ensure to rearrange if there is a -x
h moves the graph left and right (along the x axis)
k moves the graph up and down (along the y axis)
An endpoint will occur at (h,k) - reverse symbol of h value.

A y=-√x graph is reflected in the X AXIS.

A y=√-x graph is reflected in the Y AXIS.
When sketching a y=√-x graph, sometimes it will appear as a y=√h-x graph, ensure it is rearranged so that the -1 coeffi­cient is outside the brackets -> as y=√-(x-h) graph

We can restrict a function to make it one to one.
As such, a parabola can have its domain restricted so it is a half parabola - this way it is now a one-to-one function.

Functions which have different rules for different subsets of their domain are called
piecew­ise­-de­fined functions. They are also known as hybrid functions.

NOTE: when evaluating piecewise functions, ensure you pay attention to the x value, and the approp­riate domain for each rule.

NOTE: when finding inverse functions always:
- Sketch the original function using its domain, and find its range.
- Sketching the original can help decide if the inverse should be positive or negative
- Once decided, then, sketch the inverse over the same set of axes, and label y=x on the graph.
- Swap the original's domain and range to then find the inverse's .