Algebra II Ra tional Expressions & Equations

Algebra II Rational Expressions & Equations 2013-09-16 www.njctl.org

Table of Contents click on a topic to go to that section Variation Reducing Rational Expressions Multiplying Rational Expressions Dividing Rational Expressions Adding and Subtracting Rational Expressions Solving Rational Equations Graphing Rational Functions

Working with Rational  Expressions Return to Table of  Contents

Working with Rational Expressions Goals and Objectives Students will reduce rational expressions, as well as be able to add, subtract, multiply, and divide rational expressions. Students will solve rational equations. Students will graph rational functions and identify their holes, vertical asymptotes, and horizontal asymptotes.

What is a rational expression? A rational expression is a fraction but with polynomials in the numerator and denominator of the fraction. It is simply the ratio of two polynomials.

Working with Rational Expressions Why do we need this? Rational expressions are often used to simplify expressions with long polynomials in both the numerator and denominator. People are more efficient when they are  working with simple problems and situations.  Knowing how to simplify rational expressions  makes looking at graphs and other problems easier. Rational expressions and equations are often used to model more complex equations in other fields such as science and engineering. Some topics in rational expressions can be applied to are fields and forces in physics and aerodynamics.

Vocabulary Review Recall from Algebra I A monomial is a one-term expression formed by a number, a variable, or the product of numbers and variables. A polynomial is an expression that contains two or more monomials.

Vocabulary Review Continued A rational expression is an expression that can be written in the form , where a variable is in the denominator. The domain of a rational expression is all Reals excluding those that would make the denominator 0. For example, in the expression , 2 is restricted from the domain because it would create an undefined term.

Variation Inverse and  Joint Variation Return to Table of Contents

Variation Variation describes the relationship between variables. There are three types of variation: direct, inverse and joint variation. Each type describes a different relationship. We will not be discussing direct variation, as it was taught in pre-algebra and Algebra I. However, remember direct variation is used when one element increases while the other element increases. Or, visa versa, when one element decreases, the other element also decreases. Direct variation forms a linear relationship. Teacher Notes [This object is a pull tab]

Variation Inverse Variation With Inverse variation, when one element increases, the other element decreases. Or, visa versa, when one element decreases, the other element increases. Examples: As increase your amount of spending, you decrease the amount of money that you have available to you. As you pull on a rubber band to make it longer, the width of the band gets smaller. As you increase your altitude by hiking up a mountain, you will feel a decrease in the temperature.

Variation Joint Variation Joint variation is the same as direct variation, but is used when two or more elements affect what another element does. If one or both elements increase, the other element increases. Or, visa versa, when one or both elements decrease, the other element also decreases. Examples: As you either decrease the speed you drive or decrease the time you drive, you will decrease the distance you cover. As you increase the length or width of your backyard fence, you increase the area of your backyard. As you increase the radius and/or the height of a cone, you increase the volume.

Variation Using more mathematical vocabulary... Inverse: The temperature of the air varies inversely with the altitude. written as: Joint: The volume of a cone varies jointly with the square of its radius and its height. written as:

Variation Notice that in each of these variations there is an additional number whose value that does not change: This number is called the constant of variation and is denoted with a k.

Variation Steps to solving a variation problem:  1) Determine an equation based on each type of variation. Direct: y = kx   Inverse: y = k/x   Joint: y = kxz  2) Find the constant of variation(k)  3) Rewrite the equation plugging in a value for k. 4) Use the final equation to find the missing value.

Variation Example: If y varies inversely with x, and y = 10 when x = 4, find x when y = 80. Teacher

Variation Example: The volume of a square pyramid varies jointly with the area of the base (s2) and the height. If the volume is 75 when the base side is 5 and the height is 9, find the volume when the height is 12 and the base side is 4. Teacher

Variation 1 If y varies inversely with x, and y = 10 when x = -4,find y when x = 8. Teacher

Variation 2 If y varies inversely with x, and y = 3 when x = 15, find y when x = 5 Teacher

Variation 3 If y varies jointly with x and z, and y = 6 when x = 3 and z = 9,find y when x = 5 and z = 4. Teacher

Variation 4 If y varies jointly with x and z, and y = 3 when x = 4 and z = 6,find y when x = 6 and z = 8. Teacher

Reducing Rational Expressions Reducing Rational Expressions Return to Table of  Contents

Reducing Rationals A rational expression is an expression that can  be written in the form , where a variable  is in the denominator. The domain of a rational expression is all real numbers excluding those that would make the denominator 0. (This is very important when we get to solving rational equations.) For example, in the expression , 2 is  restricted from the domain.

Reducing Rationals Reducing Rational Expressions Monomials- reduce coefficients, reduce variables by subtracting  exponents. Other Polynomials- factor first, then reduce. click click click

Reducing Rationals Remember to use properties of powers or factoring, in order to simplify the rational expressions. Teacher

Reducing Rationals 5 Simplify C Teacher C B A D

Reducing Rationals 6 Simplify C Teacher B A C D

Reducing Rationals 7 Simplify B Teacher A C D B

Reducing Rationals 8 Simplify: A Teacher A C D B

Reducing Rationals 9 Simplify: B Teacher D A C B

Multiplying Rational Expressions Multiplying Rational Expressions Return to Table of  Contents

Multiplying Rational Expressions When rational expressions are multiplied, multiply the  numerators together and the denominators together. Hint: If you simplify BEFORE you multiply reducing will be easier at the end. Example: Multiplying: click

Multiplying Rational Expressions Sometimes you need to factor so you can simplify. click click

Multiplying Rational Expressions Try the following...don't forget to reduce! Teacher

Multiplying Rational Expressions 10 Simplify D Teacher A C D B

Multiplying Rational Expressions 11 Simplify B Teacher A C D B

Multiplying Rational Expressions 12 Simplify A Teacher A C D B

Multiplying Rational Expressions 13 Simplify B Teacher A C D B

Dividing Rational Expressions Dividing Rational Expressions Return to Table of  Contents

Dividing Rational Expressions Dividing Rational Expressions When dividing rational expressions, multiply by the reciprocal. Examples: click click

Dividing Rational Expressions 14 Simplify C Teacher A C B D

Dividing Rational Expressions 15 Simplify B Teacher A C D B

Dividing Rational Expressions 16 Simplify A Teacher C A B D

Dividing Rational Expressions 17 Simplify D Teacher C A D B

Dividing Rational Expressions 18 Simplify D Teacher A C B D

Adding and Subtracting Rational Expressions Return to Table of  Contents

Adding or Subtracting Rational Expressions Just like multiplication and division, when adding or  subtracting rationals, use the same rules as basic fractions. Recall: When adding and subtracting fractions, you MUST use common denominators.

Adding and Subtracting Rationals Adding and Subtracting Rational Expressions To add and subtract rational expressions they must have common  denominators. Examples: click click

Adding and Subtracting Rationals 19 Simplify B Teacher A C B D

Adding and Subtracting Rationals A 20 Simplify Teacher A C D B

Algebra II Ra tional Expressions & Equations