# RATIONALS

## 1. Simplifying Rational Expressions

Step 1) Factor numerator and denominator.

Step 2) Set up your expression to reflect the factoring.

Step 3) Cross out any factors the numerator and denominator have in common.

Step 4) Look for any other common factors you can cross out.

## 2. Multiplying and Dividing Rational Expressions

Step 1) Factor both the numerator and denominator.

Step 2) Write as one fraction.

Step 3) Simplify rational expression.

Step 4)
Multiply any remaining factors in the numerator and denominator.

For DIVISION: use reciprocal.

## 3. Adding and Subtracting Rational Expressions

To add or subtract rational expressions:

Step 1) Factor each denominator.

Step 2) Find the least common denominator (LCD) for all the denominators by multiplying together the different prime factors with the greatest exponent for each factor.

Step 3) Rewrite each fraction so it has the LCD as its denominator by multiplying each fraction by the value 1 in an appropriate form.

Step 4) Combine numerators as indicated and keep the LCD as the denominator.

Step 5) Simplify the resulting rational expression if possible.

## 4. Transformations on Rational Parent Function

f(x) = a/x-h + k

Just like every other parent function, when:

A > 1: the graph stretches
0 < a > 1: the graph compresses
a < 0: a reflection occurs

+h: translates left
-h: translates left

+k: translates up
-k: translates down

## 5. Vertical and Horizontal Asymptotes

To find asymptotes:

• set the denominator equal to zero and solve
• the zeroes (if any) are the vertical asymptotes
• everything else is the domain
• compare the degrees of the numerator and the denominator
• if the degrees are the same, then you have a horizontal asymptote at y = (numerator's leading coefficient) / (denominator's leading coefficient)
• if the denominator's degree is greater (by any margin), then you have a horizontal asymptote at y = 0 (the x-axis)
• if the numerator's degree is greater (by a margin of 1), then you have a slant asymptote which you will find by doing long division

## 6. Removable Discontinuity

• also known as the "hole"
• A point where a function is discontinuous
• found when numerator = 0
• if you plug in an x-value and get 0 over 0 then the graph will have "RD"

## 7. Solving Rational Equations

Solving Rational Equations