Algebra 2Rational and Radical

Expressions

How does asymptotes effect a graph?How can we come up with the domain and

range of rational functions?How do we interpret the graph of rational

functions?What are vertical asymptotes and how are

they used on a graph?Is it possible to simplify rational expressions?

If so, how do we simply them?What does it mean if there is a “hole” in our

curve?

Review

Rational ExpressionsA rational expression consists of a polynomial divided by a nonzero polynomial. (Examples below)

A rational function is a function defined by a formula that is a rational expression. The function is in terms of f(x). (Example below)

- Domain of a rational function is the set of all real numbers that satisfy the function except those for which the denominator is zero.

- Find the domain by determining when the denominator is zero. When the denominator equals zero the function and graph will be undefined.

- We want to find an (x) that makes the function undefined creating a “hole” in the functions graph.

- Therefore, set the denominator equal to zero and solve for x. Note: There may be more than one number that sets the denominator equal to zero.

Domain of Rational Functions

Simplifying Rational Expressions

- A rational expression is simplified if its numerator and denominator have no common factors other than 1 or -1.

Steps to simplify rational expressions:1.) Factor the numerator and the denominator completely.2.) Divide both the numerator and the denominator by any common factors.(only divide out, or cancel factors common to the numerator and denominator).

Vertical Asymptotes-The denominator also gives an (x value) where there is a vertical asymptoteVertical asymptote is a vertical line that the graph of a function approaches, BUT DOES NOT TOUCH.

a.) a graph may have zero, one, or several veritcal asymptotes

b.) no point from the function can land on an asymptote, however, it can be extremely close.

c.) MUST check the x-value that makes the denominator zero by putting it into the function and seeing if it is a vertical asymptote or hole by the solution it gives.

Does every rational function have values to exclude? (x values where the denominator equals zero)

How can you tell if an x-value (from the denominator) is going to be a “hole” or “vertical asymptote” on a graph?

Can you factor the denominator and set each factors equal to zero to solve for x?

Check for Understanding

Does every rational function have values to exclude? (x values where the denominator equals zero)

No, (x squared plus 1) has no real number values that cause the denominator to equal zero.

How can you tell if an x-value (from the denominator) is going to be a “hole” or “vertical asymptote” on a graph?

After you solve for x, substitute the x value into the equation and solve. If the solution is any number besides zero then it is a “hole”. If the solution is zero then there is a vertical asymptote at that x value.

Can you factor the denominator and set each factors equal to zero to solve for x?

Yes, but ONLY if you are looking for the domain, hole, or vertical asymptote.

Check for Understanding (answers)

Vertical Asymptote Example

The graph of the function is given in green.(if you take random x-values and substitute it in the equation the solution will give a point (x,y) that will be along one of the green curves).

The vertical asymptote is where x = -2 (from the denominator).

When x = -2, the function is undefined, therefore, there can not be an x-value that equals -2.