4A  Rectangular Hyperbolas
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 xh=0, and solve for x).
4A  Sketching Rectangular Hyperbolas
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.
4A  Rearranging Form of Rectangular Hyperbolas.
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.
7D  Determining Transformations
7C  Function Notation and Transformations
7C Combinations of Transformations
NOTE: the order in which the transformations are applied are very important
as above, use brackets to follow order of operations.
7B  Dilations and Reflections


4B  The Truncus
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 xh=0, and solve for x).
4B  Sketching a Truncus
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.
4D  The Graph of y= √x
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.
4D  Sketching a graph of y=√x
4D  Sketching a graph of y=√x
A y=√x graph is reflected in the X AXIS.
4D  Sketching a y=√x graph
A y=√x graph is reflected in the Y AXIS.
When sketching a y=√x graph, sometimes it will appear as a y=√hx graph, ensure it is rearranged so that the 1 coefficient is outside the brackets > as y=√(xh) graph


5A  Set Notation and Sets of Numbers.
5B  Relations, Domain and Range
5C  Functions
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 onetoone function.
5E  Piecewise Functions
Functions which have different rules for different subsets of their domain are called
piecewisedefined functions. They are also known as hybrid functions.
5F  Applying Function Notation
NOTE: when evaluating piecewise functions, ensure you pay attention to the x value, and the appropriate domain for each rule.
5G  Inverse functions
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 .

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