8.1: Model Inverse and Joint Variation I. Inverse Variation: Two variables x and y show inverse variation if they are related as follows: where a 0 * In this equation y is said to vary inversely with x. II. Constant of Variation: The constant in the equation above. III. Joint Variation: Joint variation occurs when a quantity with the of two or more other quantities. Examples: 1. Tell whether x and y show direct variation, inverse variation, or neither. (a) y - 3x = 0 (b) x = 3 y (c) x + y = 5 1. The variables x and y vary inversely and y = 15 when x = 3. Write an equation that relates x and y. Then find y when x = -10. 3. Moving cartons are manufactured in a variety of objects that need to be packed. The table compares the areas A of the bottom of a rectangular carton (in square inches) with the height h for four available cartons that have the same volume. A 50 100 10 150 h 4 1 10 8 (a) Write a model that gives h as a function of A. (b) Predict the height of a carton with the same volume as those in the table that has a base area of 75 square inches.
Notes 8.1 page 4. The variable z varies jointly with x and y. Also, z = 60 when x = -4 and y = 5. Find z when x = 7 and y =. 5. Write an equation for the given relationship: (a) r varies inversely with s (b) z varies jointly with x and the square root of y (c) p varies inversely with the cube of q. (d) d varies directly with the square of m and inversely with p (e) z varies jointly with u and v and inversely with the square of w.
8.: Graph Simple Rational Functions I. Rational Function: A rational function has the form where p(x) and q(x) are polynomials and q(x) 0. * The inverse variation function is a rational function. II. Parent Function for Simple Rational Functions: The graph of the parent function is a, which consists of two symmetrical parts called. Domain: Range: Any function of the form has the same,, and x-scale =.5 y-scale =.5 as the function III. Graphing Translations of Simple Rational Functions: To graph a rational function of the form, follow these steps: Step 1: Draw the asymptotes and. Step : Plot points to the left and to the right of the vertical asymptotes Step 3: Draw the two branches of the hyperbola so that they pass through the plotted points and approach the asymptotes. IV. Other Rational Functions: All rational functions of the form also have graphs that are hyperbolas. Vertical Asymptote: The line with equation (the function is undefined with the denominator is zero) Horizontal Asymptote: The line with equation
Examples: 1. Graph y 4 x Compare with the parent graph.. Graph y 3 x 1. Notes 8. page Domain: Domain: Range: Range: 3. Graph y 3x 6 x vertical asymptote: horizontal asymptote: Domain: Range: 4. Your long-distance calling plan has a fixed monthly fee of $4.95 and costs 5 cents a minute. a. Write an equation that gives your average cost C in dollars per minute m during a given month. b. Graph the function. c. Estimate when the average cost is $0.14 per minute. d. What happens to the average cost per minute as the number of minutes increases?
8.4: Multiply and Divide Rational Expressions I. Simplifying Rational Expressions: Let a, b, and c be expressions with b 0 and c 0. Then II. Multiplying Rational Expressions: Let a, b, and c be expressions with b 0 and d 0. Then III. Dividing Rational Expressions: Let a, b, c, and d be expressions with b 0, c 0, and d 0. Then Examples: Simplify: x x 3 1. x x 6. x 10x 3x 16x 5 Multiply: 3. 0x 5x x x x 3x 4 x 16 4. x 4 x 3 8 x x 4 x 16
Notes 8.4 page Divide: 5. x 4x 1 x 3x 70 5x 15 x 100 6. 3x 13x 10 (3x x) 6x
8.5: Add and Subtract Rational Expressions I. Adding or Subtracting Rational Expressions: Like Denominators: To add or subtract rational expressions with like denominators, add (or subtract) their numerators. The place the result over the common denominator. Unlike Denominators: To add or subtract rational expressions with unlike denominators, find a common denominator. Rewrite each rational expression using the common denominator. Then add (or subtract). You can always find a common denominator of two rational expressions by multiplying their denominators. However, if you use the least common multiple of the denominators, there may be less simplifying to do. II. Complex Fractions: A complex fraction is a fraction that contains a in its or. III. Simplifying Complex Fractions: Method 1: If necessary, simplify the numerator and denominator by writing each as a single fraction. Then divide the numerator by the denominator. Method : Multiply the numerator and the denominator by the least common denominator (LCD) of every fraction in the numerator and denominator. Then simplify. Examples: Perform the indicated operation. 1. 1. 5x 5x 3x 1 x 5 x 5 Find the least common multiple of the two polynomials. 3. 5x - 45 and 4x + 4x + 36
Add/Subtract: 3 4. 10x x 5x 10x 5. x 3x 15 x x 4x 5 Notes 8.5 page 6. Simplify the complex fraction (method 1): 6 x 5 1 x 3 x x 5 7. Simplify the complex fraction (method ): 3 x 5 x 3 1 x 5
8.6: Solve Rational Equations To solve a rational equation when each side is I. A Single Rational Expression: Use cross multiplying II. Not a Single Rational Expression: Multiply each side of the equation by the least common denominator of each rational expression. Examples: Solve: 1. 7 x 11 x 10. 7 x 1 5 6 x 1 3. 15 4 7 4. x 5 x 1 1 1 x 7 x 5. 5 x x 1 x x 9 x