Slide 1 / 179 Slide 2 / 179 Algebra II Rational Expressions & Equations 2015-08-15 www.njctl.org Slide 3 / 179 Slide 4 / 179 Table of Contents click on a topic to go to that section Working with Rational Expressions Joint and Inverse Variation Simplifying Rational Expressions Working with Rational Multiplying Rational Expressions Expressions Dividing Rational Expressions Adding and Subtracting Rational Expressions Solving Rational Equations Applications of Rational Equations Graphing Rational Functions Return to Table of Contents Slide 5 / 179 Slide 6 / 179 What is a rational expression? Goals and Objectives Students will simplify rational expressions, as · well as be able to add, subtract, multiply, and A rational expression is the ratio of two polynomials. It is divide rational expressions. written as a fraction with polynomial expressions in the Students will solve rational equations and use · numerator and denominator. them in applications. Students will graph rational functions and · identify their holes, vertical asymptotes, and horizontal asymptotes.
Slide 7 / 179 Slide 8 / 179 Why do we need this? Rational expressions are often used to simplify expressions with long polynomials in both the numerator and denominator. Since it is more efficient to work 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 fields such as science and engineering. Rational expressions are applicable in working with forces and fields in physics and aerodynamics. Slide 9 / 179 Slide 10 / 179 Inverse and Joint Variation Return to Table of Contents Slide 11 / 179 Slide 11 (Answer) / 179 Variation Variation Variation describes the relationship between variables. Variation describes the relationship between variables. We will not be discussing direct There are three types of variation: There are three types of variation: variation, as it was taught in pre- Teacher Notes algebra and Algebra I. However, direct, direct, direct variation is used when one element increases while the other inverse and inverse and element increases. Or, vice versa, joint variation. joint variation. when one element decreases, the other element also decreases. Each type describes a different relationship. Each type describes a different relationship. Direct variation forms a linear relationship. [This object is a pull tab]
Slide 12 / 179 Slide 13 / 179 Joint Variation Inverse Variation Joint variation is the same as direct variation, but is used when two or With Inverse variation, when one element increases, the other more elements affect what another element does. If one or both element decreases. Or, vice versa, when one element elements increase, the other element increases. Or, vice versa, when decreases, the other element increases. one or both elements decrease, the other element also decreases. Examples: Examples: As you increase As you either As you increase As you pull on a As you increase As you increase the radius and/ decrease the speed rubber band to your altitude by the length or your amount of or the height of you drive or make it longer, hiking up a width of your spending, you a cone, you decrease the time the width of the mountain, you backyard fence, decrease the increase the you drive, you will band gets will feel a you increase the amount of money volume. decrease the smaller. decrease in the area of your available to you. distance you cover. temperature. backyard. Slide 14 / 179 Slide 15 / 179 Variation Variation Using more mathematical vocabulary... Notice that in each of these variations there is an additional Inverse variation: The temperature of the air varies number whose value does not change: inversely with the altitude. written as: Joint variation: The volume of a cone varies jointly This number is called the constant of variation and with the square of its radius and its height. is denoted by k . written as: Slide 16 / 179 Slide 17 / 179 Variation Variation Example: Steps to solving a variation problem: If y varies inversely with x, and y = 10 when x = 4, find x when y = 80. 1) Determine an equation based on each type of variation. Inverse: y = k / x Joint: y = kxz 2) Find the constant of variation (k) 3) Rewrite the equation substituting a value for k. 4) Use the final equation to find the missing value.
Slide 17 (Answer) / 179 Slide 18 / 179 Variation Variation Example: Example: The volume of a square pyramid varies jointly with the area of the base (s 2 ) and the height. If the volume is 75 when If y varies inversely with x, and y = 10 when x = 4, find x when y = 80. the base side is 5 and the height is 9, find the volume when the height is 12 and the base side is 4. Answer [This object is a pull tab] Slide 18 (Answer) / 179 Slide 19 / 179 Variation 1 If y varies inversely with x, and y = 10 when x = -4, find y when x = 8. Example: The volume of a square pyramid varies jointly with the area of the base (s 2 ) 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. Answer [This object is a pull tab] Slide 19 (Answer) / 179 Slide 20 / 179 2 If y varies inversely with x, and y = 3 when x = 15, find y when x = 5.
Slide 20 (Answer) / 179 Slide 21 / 179 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. Slide 21 (Answer) / 179 Slide 22 / 179 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. Slide 22 (Answer) / 179 Slide 23 / 179 Simplifying Rational Expressions Return to Table of Contents
Slide 24 / 179 Slide 25 / 179 Simplifying 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 solving rational equations.) For example, in the expression , 2 and -2 are restricted from the domain. Slide 26 / 179 Slide 26 (Answer) / 179 Slide 27 / 179 Slide 27 (Answer) / 179 Simplifying Rationals Simplifying Rationals Remember to use properties of exponents and/or Remember to use properties of exponents and/or factoring to simplify the rational expressions. factoring to simplify the rational expressions. Answer [This object is a pull tab]
Slide 28 / 179 Slide 28 (Answer) / 179 5 Simplify 5 Simplify Answer C C C A B A B D D [This object is a pull tab] Slide 29 / 179 Slide 29 (Answer) / 179 6 Simplify 6 Simplify Answer B B B A C D A C [This object is a pull tab] D Slide 30 / 179 Slide 30 (Answer) / 179 7 Simplify 7 Simplify Answer B [This object is a pull tab] A C D A C D B B
Slide 31 / 179 Slide 31 (Answer) / 179 Slide 32 / 179 Slide 32 (Answer) / 179 Slide 33 / 179 Slide 34 / 179 Multiplying Rational Expressions Return to Table of Contents
Slide 34 (Answer) / 179 Slide 35 / 179 Slide 36 / 179 Slide 37 / 179 Multiply Slide 37 (Answer) / 179 Slide 38 / 179 Multiply Answer [This object is a pull tab]
Slide 38 (Answer) / 179 Slide 39 / 179 10 Simplify A C B D Slide 39 (Answer) / 179 Slide 40 / 179 10 Simplify A C Answer D D B [This object is a pull tab] Slide 40 (Answer) / 179 Slide 41 / 179
Slide 41 (Answer) / 179 Slide 42 / 179 Slide 42 (Answer) / 179 Slide 43 / 179 Dividing Rational Expressions Return to Table of Contents Slide 44 / 179 Slide 45 / 179 14 Simplify A C B D
Slide 45 (Answer) / 179 Slide 46 / 179 14 Simplify 15 Simplify A C Answer A C C B D D B [This object is a pull tab] Slide 46 (Answer) / 179 Slide 47 / 179 Slide 47 (Answer) / 179 Slide 48 / 179
Slide 48 (Answer) / 179 Slide 49 / 179 18 Simplify A C D B Slide 49 (Answer) / 179 Slide 50 / 179 18 Simplify Adding and Subtracting Answer D Rational Expressions A C [This object is a pull tab] B D Return to Table of Contents Slide 51 / 179 Slide 52 / 179 Adding and Subtracting Rational Expressions Just as in 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.
Slide 53 / 179 Slide 53 (Answer) / 179 Slide 54 / 179 Slide 54 (Answer) / 179 Slide 55 / 179 Slide 55 (Answer) / 179
Slide 56 / 179 Slide 57 / 179 Adding and Subtracting Rational Expressions Adding and Subtracting Rational Expressions To add and subtract rational expressions they must have common denominators. Identify the LCD and rewrite the rational expressions with the same denominator. Step 1: Find LCD Example: No common denominator. Step 2: Multiply each term by an expression equal to 1 to obtain LCD for each term. x 2 is the least common denominator.(LCD) Step 3: Add or subtract numerators Multiply by an expression equal to 1. (Multiply numerator and denominator by the same quantity.) Step 4: Simplify Then Add. Simplify if possible. Slide 58 / 179 Slide 59 / 179 Example Example Solve: Solve: The denominators are additive inverses. Step 1: LCD = ( x + 2)( x - 2) Step 1: Step 2: Step 2: Step 3: Step 3: Step 4: Step 4: Slide 60 / 179 Slide 61 / 179
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