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Chapter 9

Chapter 9. Rational Functions. In this chapter you should …. Learn to use inverse variation and the graphs of inverse variations to solve real-world problems. Learn to identify properties of rational functions. Learn to simplify rational expressions and to solve rational equations.

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Chapter 9

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  1. Chapter 9 Rational Functions

  2. In this chapter you should … • Learn to use inverse variation and the graphs of inverse variations to solve real-world problems. • Learn to identify properties of rational functions. • Learn to simplify rational expressions and to solve rational equations.

  3. What you’ll learn … To write and interpret direct variation equations 2-3 Direct Variation • 1.05 Model and solve problems using direct, inverse, combined and joint variation.

  4. This is a graph of direct variation.  If the value of x is increased, then y increases as well.Both variables change in the same manner.  If x decreases, so does the value of y.  We say that y varies directly as the value of x. 

  5. Definition: Y varies directly as x means that y = kx where k is the constant of variation. (see any similarities to y = mx + b?) Another way of writing this is k =

  6. Example 1a Identifying Direct Variation from a Table For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. k = _______ k = _______ Equation _________________ Equation _________________

  7. Example 1b Identifying Direct Variation from a Table For each function, determine whether y varies directly with x. If so, find the constant of variation and write the equation. k = _______ k = _______ Equation _________________ Equation _________________

  8. Example 4a Using a Proportion Suppose y varies directly with x, and x = 27 when y = -51. Find x when y = -17.

  9. Example 4b Using a Proportion Suppose y varies directly with x, and x = 3 when y = 4. Find y when x = 6.

  10. Example 4c Using a Proportion Suppose y varies directly with x, and x = -3 when y = 10. Find x when y = 2.

  11. What you’ll learn … To use inverse variation To use combined variation 9-1 Inverse Variation • 1.05 Model and solve problems using direct, inverse, combined and joint variation.

  12. In an inverse variation, the values of the two variables change in an opposite manner - as one value increases, the other decreases.  Inverse variation:  when one variable increases,the other variable decreases.

  13. Inverse Variation When two quantities vary inversely, one quantity increases as the other decreases, and vice versa. Generalizing, we obtain the following statement. An inverse variation between 2 variables, y and x, is a relationship that is expressed as: where the variable k is called the constant of proportionality. As with the direct variation problems, the k value needs to be found using the first set of data.

  14. Example 2a Identifying Direct and Inverse Variation Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations.

  15. Example 2a Identifying Direct and Inverse Variation Is the relationship between the variables in each table a direct variation, an inverse variation, or neither? Write functions to model the direct and inverse variations.

  16. Example 3 Real World Connection Zoology. Heart rates and life spans of most mammals are inversely related. Us the data to write a function that models this inverse variation. Use your function to estimate the average life span of a cat with a heart rate of 126 beats / min.

  17. A combined variation combines direct and inverse variation in more complicated relationships. k x3 kxy w kx wy

  18. Example 5a Finding a Formula e The volume of a regular tetrahedron varies directly as the cube of the length of an edge. The volume of a regular tetrahedron with edge length 3 is . Find the formula for the volume of a regular tetrahedron. e 9 √ 2 4

  19. Example 5b Finding a Formula e e The volume of a square pyramid with congruent edges varies directly as the cube of the length of an edge. The volume of a square pyramid with edge length 4 is . Find the formula for the volume of a square pyramid with congruent edges. e 32 √ 2 3

  20. What you’ll learn … To identify properties of rational functions To graph rational functions 9-3 Rational Functions and Their Graphs • 2.05 Use rational equations to model and solve problems; justify results. • Solve using tables, graphs, and algebraic properties. • Interpret the constants and coefficients in the context of the problem. • Identify the asymptotes and intercepts graphically and algebraically.

  21. Definition Rational Function A rational function f(x) is a function that can be written as where P(x) and Q(x) are polynomial functions and Q(x) ≠ 0. P(x) Q(x) f(x) =

  22. Examples of Rational Functions -2x x2 + 1 y = In this graph, there is no value of x that makes the denominator 0. The graph is continuous because it has no jumps, breaks, or holes in it. It can be drawn with a pencil that never leaves the paper.

  23. Examples of Rational Functions In this graph, x cannot be 4 or -4 because then the denominator would equal 0. 1 x2 - 16 y =

  24. Examples of Rational Functions (x+2)(x-1) x - 1 In this graph, x cannot equal 1 or the denominator would equal 0. y =

  25. Point of Discontinuity A function is said to have a point of discontinuity at x = a or the graph of the function has a hole at x = a, if the original function is undefined for x = a, whereas the related rational expression of the function in simplest form is defined for x = a. 

  26. Example of Point of Discontinuity • Consider a function . • This function is undefined for x = 2. But the simplified rational expression of this function, x + 3 which is obtained by canceling (x - 2) both in the numerator and the denominator is defined at x = 2. Thus we can say that the function f(x) has a point of discontinuity at x = 2.

  27. Example 1b Finding Points of Discontinuity 1 x2 - 16 x2 - 1 x2 + 3

  28. Vertical Asymptotes • An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters. • If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd. • Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.

  29. Example 1a Finding Points of Discontinuity 1 x2 + 2x +1 -x + 1 x2 +1

  30. Example 2a Finding Vertical Asymptotes (x – 2) (x – 1) x - 2 x + 1 (x – 2)(x – 3)

  31. Example 2b Finding Vertical Asymptotes x – 2 (x - 1)(x + 3) (x – 3)(x + 4) (x – 3)(x – 3)(x+4)

  32. Horizontal Asymptotes • The graph of a rational function has at most one HA. • The graph of a rational function has a HA at y=0 if the degree of the denominator is greater than the degree of the numerator . • If the degrees of the numerator and the denominator are =, then the graph has a HA at y = , a is the coefficient of the term of the highest degree in the numerator and b is the coefficient of the term of the highest degree in the denominator. • If the degree of the numerator is greater than the degree of the denominator, then the graph has no HA a b

  33. Example 4a Sketching Graphs of HA x + 2 (x+3)(x-4) y =

  34. Example 4b Sketching Graphs of HA x + 3 (x-1)(x-5) y =

  35. The CD-ROMs for a computer game can be manufactured for $.25 each. The development cost is $124,000. The first 100 discs are samples and will not be sold. Write a function for the average cost of a salable disc. Graph the function. What is the average cost if 2000 discs are produced? If 12,800 discs are produced? Example 5 Real World Connection

  36. What you’ll learn … To simplify rational expression To multiply and divide rational expressions 9-4 Rational Expressions • 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

  37. A rational expression is in its simplest form when its numerator and denominator are polynomials that have no common divisors.

  38. -27x3y 9x4y Example 1a Simplifying Rational Expressions x2 + 10x + 25 x2 + 9x + 20

  39. 2x2 – 3x - 2 x2 – 5x + 6 Example 1b Simplifying Rational Expressions -6 – 3x x2 - 6x + 8

  40. Example 2 Real World Connection SA = 2rh + 2r2 Architecture One factor in designing a structure is the need to maximize the volume (space for working) for a given surface area (material needed for construction). Compare the ratio of the volume to surface area of a cylinder with radius r and height r to a cylinder with radius r and height 2r.

  41. Multiplying Rational Expressions Simply Put: The rule for multiplying algebraic fractions is the same as the rule for multiplying numerical fractions. Multiply the tops (numerators) AND multiply the bottoms (denominators). If possible, reduce (cancel) BEFORE you multiply the tops and bottoms!(It's easier than simplifying at the end!)

  42. Example 3a Multiplying Rational Expressions

  43. Example 3b Multiplying Rational Expressions

  44. Dividing Rational Expressions Simply Put:The rule for dividing algebraic fractions is the same as the rule for dividing numerical fractions. Change the division sign to multiplication, flip the 2nd fraction ONLY, and then follow the steps for "multiplying rational expressions".

  45. Example 4a Dividing Rational Expressions

  46. Example 4b Dividing Rational Expressions

  47. What you’ll learn … To add and subtract rational expressions To simplify complex fractions 9-5 Adding and Subtracting Rational Expressions • 1.03 Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

  48. The Basic RULE for Adding and Subtracting Fractions: • Get a Common Denominator! (the smallest number that both denominators can divide into without remainders.)  • With each fraction, whatever is multiplied times the bottom must ALSO be multiplied times the top. • Do not add the common denominators.  Add only the numerators (tops).

  49. 2 5 3 3 Adding and Subtracting Fractions with Like Denominators 4 3 7 7 + -

  50. 2 5 x + 3 x + 3 Adding Expressions with Like Denominators y y + 3 y – 5 y – 5 + +

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