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 = x² is a power function with a = 1 and b = 2.

Symmetry with respect to the y-axis

 The graph of y = f(x) is symmetric with respect to the y-axis 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 y-axis. They will all be even functions.

Now let's graph a power function with an odd exponent. f(x) = x³. We will use a T-table.

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 x-axis 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 x-axis.

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 x-axis.

b) the graph of y = f(-x) is a reflection of f across the y-axis.

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.