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Unit 7—Rational Functions Rational Expressions • Quotient of 2 polynomials 3x 2 y 2x 5 2 x 2 3x 1 or or or x 9 3 15 xy y 3 y 1 p so ,q 0 q Things to Consider Graphing Rational Functions 1. Factor 2. Determine where discontinuities would occur in the graph 3. Graph any asymptotes on the graph and pick points on both sides to locate the branches of the function • If factors cancel, then you have a point of discontinuity (hole) • If factors remain in the denominator, you have a vertical asymptote(s) Horizontal Asymptotes 1. If the degree of the numerator is bigger than the degree of the denominator, then there is NO H.A. (top-heavy) 2. If the degree of numerator is smaller than the degree of the denominator, then the HA is at y=0 (bottom heavy) 3. If the degree of numerator is equal to degree of the denominator, then the HA is equal to the leading coefficients (equal weight) Graphing Rational Functions x 2 7 x 12 y 2 x 9 x 20 Graphing Rational Functions 3x 13x 10 y x 5 2 Graphing Rational Functions x 2x 3 y 2 x x2 2 Graphing Rational Functions x y 2 x 2x 3 Graphing Rational Functions x 9 y x 3 2 Graphing Rational Functions x 9 y 2 x 1 2 To simplify a rational expression • Look for common factors 27 x 3 y ex. 9x4 y ( x 2) ex. ( x 2)( x 1) 4x2 9 ex. 4 x 2 12 x 9 x3 ex. x 3 To simplify a rational expression • Look for common factors x2 ex. 2 x 6 3x ex. 4 x 2 5x 6 x5 ex. 25 x 2 Multiply Rational Expressions • Factor, Reduce common factors first, then multiply x2 9 x2 4x 4 ex. 2 x 4 x3 ( x 3) 2 x4 ex. 2 x 7 x 12 x 3 Multiply Rational Expressions • Factor, Reduce common factors first, then multiply 4ab3 b 2 16 ex. 2 4b b 8a x 3 27 x 2 25 ex. 2 x 9 x5 Dividing Rational Expressions • Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply 3x 6 x 2 5x 6 ex. 12 x 24 3x 2 12 x2 3x ex. 2 2 x 2x 1 x 1 Dividing Rational Expressions • Rewrite as Multiplication by reciprocal of 2nd fraction, factor, Reduce common factors, then multiply 3x 2 9 x x 2 9 ex. x2 4x 8 4 x 3 16 x 2 ex. 4 2 3y 9y Dividing Rational Expressions x 3 25 x 2 x 2 2 x 2 5 x ex. 2 2 x 6x 5 4x 7x 7 Dividing Rational Expressions ax ay bx by ax ay bx by ex. ax ay bx by ax ay bx by Dividing Rational Expressions x 9 x 14 2 x 6x 5 2 x 8x 7 2 x 7 x 10 2 Dividing Rational Expressions x 1 x 1 x 2 x 1 x 1 x 1 Adding and Subtracting Rational Expressions • To add fractions, you must have a common denominator • To determine the LCD, list any common factor that occurs in two or more of the denominators only once in the LCD and then include all other factors that are not common. Adding and Subtracting Rational Expressions x 1 x ex. x4 x4 2x 5 x 4 ex. 2 2 x 1 x 1 Adding and Subtracting Rational Expressions 6x 4x ex. 3x 1 2 x 5 5x 3 4x ex. 2 3 x x 9 Adding and Subtracting Rational Expressions 4 3 ex. 2 2 x 16 x 8 x 16 2x 4 x4 ex. 2 x x x( x 1)( x 1) Adding and Subtracting Rational Expressions 1 5x 3 ex. 2 2x x 1 x 1 7y 8 3 ex. 2 y y 2 y y2 Adding and Subtracting Rational Expressions 2 x 10 5 x ex. 2 x 25 25 x 2 3y 2 7 ex. 2 2 y 5 y 24 y 4 y 32 Complex Rational Expressions 1 1 x y 1 1 x Complex Rational Expressions x4 y2 2 x 1 1 x Complex Rational Expressions 35 m m 12 63 m m2 Complex Rational Expressions 1 1 x y 1 1 x y Complex Rational Expressions x x 3 x x 6 Solving Rational Equations 1. 2. 3. 4. Factor all denominators Multiply both sides of equation by LCD Solve Eliminate any solution that would make the denominator zero 5. Check remaining solutions Solving Rational Equations 5 4 8 2 x x 3 x 3x Solving Rational Equations x 1 4 2 5 x Solving Rational Equations x2 x 4 2 x 3 x 10 x 5 Solving Rational Equations y 1 2 y 3 y 3 Solving Rational Equations 3 2 m 1 m 3 Solving Rational Equations 50 50 4 x x2 x Solving Rational Equations 3 2y 5 2 y2 4 y y2 Solving Rational Inequalities 1. State the excluded values 2. Solve the related equation 3. Use those values on a number line and test the values Solving Rational Inequalities x2 x 2( x 3) x 3 Solving Rational Inequalities 4 1 c2 Graphing Rational Functions Possible Graphs: Direct and Inverse Variation • Direct Variation can be expressed in the form y=kx • K is the constant of variation • Equation of variation—equation representing the relationship between the variable but substitute the value of k Ex. Y=2x (if k=2) Direct and Inverse Variation y varies directly w ith x. When y 12, x 3. Find y when x 10 Direct and Inverse Variation y varies directly w ith x. When y 25, x 5. Find x when y 1 Direct and Inverse Variation Q varies directly w ith square of p. When q 8, p 3. Find q when p 4 Direct and Inverse Variation 1 L varies directly w ith cube of m. When L , m 2. 2 Find L when m 3 Inverse Variation • Expressed as y=k/x Direct and Inverse Variation 2 y varies inversely with x. When y 3, x . 5 3 Find y when x 4 Direct and Inverse Variation 1 y varies inversely with the square of x. When y 2, x . 2 Find y when x 4 Direct and Inverse Variation y varies inversely with x. When y 6, x -3. Find x when y 8 Joint Variation • When one quantity varies directly with the product of two or more other quantities • Combination Variation—when one quantity varies directly with another quantity and inversely with the other quantity Joint or Combination Variation y varies directly w ith x and the square of z. When y 4, x 2 and z 3. Find y when x 8and z 6 Joint or Combination Variation y varies directly w ith x and inversely with the square of z. When y 20, x 4 and z 1. Find y when x 5 and z 6