# Combination of Function: Evaluation of Division

Example:

Evaluate when and

Solution:

 Evaluate the function Rewrite the expression Replace x with -3 in each function Simplify Division by zero is undefined is undefined

is undefined.

# Difference Quotient: Quadratic Function

Example:  Find the difference quotient for

The Difference Quotient:

Solution:

 The Difference Quotient Formula Write the difference quotient for the given function = Apply the exponent and use the distributive property = Multiply = Combine the like terms = Combine the like terms.  Only terms with h should remain = Divide h into each term = Cancel the common h from each term =

# Difference Quotient: Rational Function

Example:  Find the difference quotient for

The Difference Quotient:

Solution:

 The Difference Quotient Formula Write the difference quotient for the given function = Use the distributive property = Simplify the complex fraction by multiplying the numerator and denominator by the common denominator = Distribute the common denominator to each fraction in the numerator. = Cancel the common factor = Multiply the expression in the numerator = Combine like terms = Cancel a common h from the numerator and denominator =

# Difference Quotient: Rational Function

Example:  Find the difference quotient for

The Difference Quotient:

Solution:

 The Difference Quotient Formula Write the difference quotient for the given function = Simplify the complex fraction by multiplying the numerator and denominator by the common denominator = Distribute the common denominator to each fraction in the numerator. == Cancel the common factor == Simplify by squaring the binomial === Simplify by combining like terms and distributing the negative == Combine like terms = Cancel a common h from the numerator and denominator =

The difference quotient for is

# Graphing an Exponential Equation by Transformations

Example: For the function below.  Graph using transformations.  Find the y-intercept.  State the horizontal asymptote and the domain and range.

First we must examine the base function

Graph using plotting points.  We can use the standard set of x-values to find ordered pairs.

xy
-22^(-2)=1/4
-12^(-1)=1/2
02^0=1
12^1=2
22^2=4

The graph below shows the points plotted and the line that connects them.  This graph has a horizontal asymptote at y=0.  The domain is   and the range is

Analyze the transformations.

The +2 in the exponent shifts the graph left 2 units.

The – in the front of the base reflects the graph over the x-axis.

The +2 next to the base shifts the graph and the horizontal asymptote up two units.

You can see the graph after the transformations.

The horizontal asymptote is y=2.  The domain is   and the range is

To find the y-intercept we let x=0.

Thus the y-intercept is (0,-2)

# Finding the Distance

The Distance Formula

Suppose A is and B is

The distance between points A and B is given by the following formula.

Example: Find the distance between points A and B.

Point A is and point B is