advertisement

Rational Expressions & Equations/ Reciprocals Math 20 – Pre-Calculus Chapter 6 Name:__________________ Class:___________________ 1 Algebra and Number Specific Outcomes Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). General Outcome: Develop algebraic reasoning and number sense. Achievement Indicators: The following set of indicators may be used to determine whether students have met the corresponding specific outcome 4.1 Compare the strategies for writing equivalent forms of rational expressions to the strategies for writing equivalent forms of rational numbers. 4.2 Explain why a given value is non-permissible for a given rational expression. 4.3 Determine the non-permissible values for a rational expression. 4.4 Determine a rational expression that is equivalent to a given rational expression by multiplying the numerator and denominator by the same factor (limited to a monomial or a binomial), and state the non-permissible values of the equivalent rational expression. 4.5 Simplify a rational expression. 4.6 Explain why the non-permissible values of a given rational expression and its simplified form are the same. 4.7 Identify and correct errors in a simplification of a rational expression, and explain the 2.1 reasoning. 5.1 Compare the strategies for performing a given operation on rational expressions to the strategies for performing the same operation on rational numbers. 5.2 Determine the non-permissible values when performing operations on rational expressions. 5.3 Determine, in simplified form, the sum or difference of rational expressions with the same denominator. 5.4 Determine, in simplified form, the sum or difference of rational expressions in which the denominators are not the same and which may or may not contain common factors. 5.5 Determine, in simplified form, the product or quotient of rational expressions. 5.6 Simplify an expression that involves two or more operations on rational expressions. (It is intended that the rational equations be those that can be simplified to linear and quadratic equations.) 6.1 Determine the non-permissible values for the variable in a rational equation. 6.2 Determine the solution to a rational equation algebraically, and explain the process used to solve the equation. 6.3 Explain why a value obtained in solving a rational equation may not be a solution of the equation. 6.4 Solve problems by modelling a situation using a rational equation. 2 Big Ideas: By the end of this unit you should be able to . . . FACTOR rational expressions in order to simplify them Determine for which value(s) of x an expression is undefined (will usually involve factoring) Add and subtract rational expressions which usually involves finding common denominators and making equivalent expressions Multiply and divide rational expressions which usually involves factoring, simplifying and then multiplying Solve rational EQUATIONS Create a system of rational equations from a word problem and solve 3 6.1 Rational Expressions Review of Fraction Basics Fractions and “0” Simplifying Rational Expressions / Non-Permissible Values Rational Expression – Non-permissible values – Example: For each rational expression, determine all the non-permissible values. 5x 5x x2 5x 6 a) b) c) 4 xy 2 x( x 5) x 2 36 4 Rational expressions can be simplified by: Example: Simplify each rational expression. State the value(s) of any restrictions. 5x 5x 12 3x a) b) c) 2 4 xy x( x 5) ( x 5)( x 4) e) x2 5x 6 x 2 36 f) x 2 81 9 x g) Homework Pg. 317 (#1-4, 6-8) 5 3x 2 8 x 3 3x 2 15 x 18 6.2 Multiplying and Dividing Rational Expressions Review of Steps to Multiplying or Dividing Fractions Simplify each of the following. (a) 5 3 8 7 (b) 4 2 7 3 (c) 2 2 3 3 (d) 8 3 9 (e) 5 15 6 16 (f) 3 4 6 5 5 (g) 3 5 8 5 4 6 9 12 6 To Multiply Rational Expressions Example: Multiply and write your solution in simplest form. Identify all non-permissible values. a) 3x3 y 3 y2 6x b) x 1 x2 2 x 5x 6 x 5x 4 c) x 2 16 x 64 x 4 x2 4x x 8 d) 2 x 2 x 1 4 x 2 28 x 48 x 2 2 x 3 2 x 2 13x 7 7 2 To Divide Rational Expressions The Non-Permissible Values in Division of Rational Expressions For example, in: A A C B B D C D the non-permissible values are B 0, D 0 and C 0 Example: Divide and write your solution in simplest form. Identify all non-permissible values. 5 x 4 25 x 2 a) 2 6 x 2 3x 18 x 2 3x 2 c) 2 x 6 x 9 x 2 8 x 15 x 2 2x 4 b) 4x 5 4x 5 8 Example: Simplify the following expression and identify any non-permissible values 4 x 20 x 4 x6 2 x 6 x 25 2 x 8 Homework pg. 327 (#1,2, 4, 7,8,12) 9 6.3 Adding and Subtracting Rational Expressions Review of Steps to Adding or Subtracting Fractions Simplify. Express all answers in simplest form. (a) 3 2 7 7 (b) 2 5 3 8 (c) 3 4 7 21 (d) 4 1 9 4 (e) 3 2 5 4 3 12 (f) 35 5 1 36 18 3 10 To Add or Subtract Rational Expressions Case 1: Denominators are the same Case 2: Denominators are different Example: Determine each sum or difference. Identify all non-permissible values. a) 2 x 4 7 x 10 x2 9 x2 9 b) 10a 5 3a 2 ab ab Example: Identify the lowest common denominator for each group of rational expressions. a) 1 1 1 , , 2 x 3x y b) 1 1 1 , , x 5 x 5 x 4 x 4 x 7 11 c) 1 1 , 2 x 6 x 9 x 8 x 15 2 Example: Determine each sum or difference. Identify all non-permissible values.. 2x 4 3 a) xy x 2 x y 2 20 y 2 b) 2 y 4 y2 1 1 2 c) 4 x x 8 x 16 1 x d) 1 x x 1 Homework pg. 336 (#1,3-10) 12 6.4 Rational Equations Rational Equation – Steps to Solving a Rational Equation: Example: Solve each of the following equations. Identify all non-permissible values. (a) 5 3 9 2x 4 4x (b) 13 x 2 x 1 x3 x2 (c) 5x 3x 8 2 x4 x4 (d) 14 x x 1 5 x 1 x 4 x 2 3x 4 Example: Colin rows his boat 24 km downstream and back to where he began. When the average speed of the current is 2 km/h, Colin can complete the journey in 9 hrs. What is Colin’s average rowing speed in still water? Downstream Distance (km) Average speed (km/h) Time (h) 15 Upstream Example: Katie mows the lawn in 40 min. When Will and Katie work together, they can mow the lawn in 24 min. How long would it take Will to mow the lawn on his own? Time Fraction of lawn mowed by Katie Fraction of lawn moved by Will 1 2 24 Homework pg. 348 (#1-4, 12-14) 16 Fraction of lawn mowed working together 7.4 Reciprocal Functions INVESTIGATION: For each given value of f(x), determine the corresponding value 1 of : f x af f x -10 000 -10 -1 -0.0001 0 0.0001 1 10 10000 Observations . . . . 17 1 f x IN GENERAL: af For any function, y f x , the graph of the reciprocal function, y 1 , can be found f x af using the following general rules: Where the value of the original function is zero, the value of the reciprocal function is _________________, and a vertical asymptote exists. Where the value of the original function is a very small positive number, the value of the reciprocal function is _____________________________. Where the value of the original function is a very small negative number, the value of the reciprocal function is _____________________________. Where the value of the original function is a very large positive number, the value of the reciprocal function is _____________________________. Where the value of the original function is a very large negative number, the value of the reciprocal function is _____________________________. Where the value of the original function is 1, the value of the reciprocal function is _______. Where the value of the original function is –1, the value of the reciprocal function is _______. These points, where the original function and reciprocal function have the same unchanged value, are called 18 Reciprocal function – Vertical Asymptote – Horizontal Asymptote – Invariant Point(s) – af Example: The graph of y f x is shown below. Use this graph to draw the graph of 1 y . f x af 19 af Example: A function is defined by the expression f x x 2. (a) Find the equation of its reciprocal function, y 1 . f x af af (b) Sketch the graph of f x x 2. Then use the graph of y = f(x) to sketch the graph 1 of y on the same grid. f ( x) (c) For the reciprocal function, y i. Domain: 1 , determine: f x iii. Vertical Asymptote: af ii. Range: iv. Horizontal Asymptote: 20 Example: A function is defined by the equation y x 2 4. Use the graph of this 1 function to sketch the graph of y 2 then analyze this reciprocal function by stating x 4 the domain, range, zeros, asymptotes, and the coordinates of any max/min points. i. Domain: ii. Range: iii. Zeros iv. max/mins: v. Vertical Asymptote: vi. Horizontal Asymptote: Homework: pg. 403 (#1-9) 21 22