Power Functions and Transformations
A Power Function is a function of the form y = ax^{b} where a and b are real numbers and b
> 0.
We have already graphed some power functions. y = x^{2} is a power function with a
= 1 and b = 2.
Symmetry with respect to the yaxis
The graph of y = f(x) is symmetric with respect to the yaxis if oe
f(x) = f(x) for every x in the domain. Such a function is called an even
function.
All power functions with even integer exponents will be symmetric with respect to the yaxis. They will all
be even functions.
Now let's graph a power function with an odd exponent. f(x) = x^{3}. We will use a Ttable.
Symmetry with respect to the origin
The graph of y = f(x) is symmetric with respect to the origin if
f(x) =  f(x) for every x in the domain. Such a function is called an odd function.
All power functions with odd integer powers will be symmetric to the origin. They will all be odd
functions.
Root Functions A root function is a function of the form y = ax^{1/n} or
where n is an integer greater than or equal to 2.
Transformations of Graphs
I. Vertical Shifts
In general, given any function y = f(x), y = f(x) + k for k > 0 will result in a shift upwards of k units.
y = f(x)  k will result in a shift downwards of k units.
II. Horizontal Shifts
In general, given any function y = f(x), y = f(x  k) for k > 0 will result in a shift to the right k units.
y = f(x + k) will result in a shift to the left k units.
III.Compressions and Stretches
In general, given any function y = f(x), y = k.f(x) for k > 1 will result in a graph that is shaped like
y = f(x), but stretched away from the xaxis by a factor of k.
In general, given any function y = f(x), y = k . f(x) for 0 < k < 1 will result in a graph that is shaped like y
= f(x), but compressed closer to the xaxis.
IV. Reflexions across an axis
In general for a function y = f(x),
a) the graph of y = f(x) is a reflection of f across the xaxis.
b) the graph of y = f(x) is a reflection of f across the yaxis.
V. Combining Transformations
Some functions involve shifts in two directions as well as a reflection and either a stretch or a
compression.
Equations Involving Roots; Radical Equations
These equations will involve square roots, cube roots, fourth
roots, etc. We need a property for solving this type of equation.
Property for Solving Radical Equations
If P and Q are algebraic expressions, then every solution of
P = Q is also a solution of (P)^{n} = (Q)^{n} for n, any positive integer.
NOTE: This property allows us to raise both sides of an equation to a positive integral power. It does,
however, also tell us that we may get a solution that works for (P)^{n} = (Q)^{n}
, but may not work for P = Q
Steps for Solving Radical Equations Analytically:
1. Isolate the radical, if possible.
2. Raise both sides of the equation to a power which eliminates the radical(s).
3. If a radical remains, repeat steps 1 and 2.
4. Solve the resulting equation.
5. Check solutions in the equation. Only those that satisfy the original equation are actual original
solutions.
