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INTRO TO RATIONAL FUNCTIONS Light It Up is the newest attraction at the Springfield Fair. In this game, a laser pointer slides along a string at a height of 1.5 m from the ground. A 10-cm platform is positioned 25 cm from a wall, and a mirror is placed on top of the platform, parallel with the ground. The object is to slide the laser pointer so that when the beam is reflected off the mirror, it will hit a target that is 1 m off the ground.A player is given just one attempt to hit the target. If she does, she then tries to hit a target that is 3 m off the ground. And if she hits that one, too, she tries for a target thatis 10 m off the ground. If she hits all three, she wins the Grand Prize: a DVD player and a DVD copy of Itchy and Scratchy’s Greatest Hits.Bart and Lisa have played the game several times, yet they haven’t been able to hit the target. Lisa is sure that there must be a mathematical equation that would allow them to determine the correct position for the laser pointer. There issuch an equation, it is a special kind known as a rational function Before you can help Lisa and Bart win this game, you must first understand the behavior of these functions CLIMIBING THE WALL 1. TheClimbing Wall is a popular event at the Springfield Fair. The wall is 100 meters high. Bart can climb at a rate of 8 meters per minute, but other participants climb at different speeds. a.Complete the table to show that the time it takes to reach the top of wall depends on the climber’s speed . b. Determine an equation that describes the table of data c. Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in negative values and solve for cost) d. Describe the domain and range. Are there any limiting values of the function? Activity taken from © 2008National Council of Teachers of Mathematics http://illuminations.nctm.org TRIP TO THE FAIR 2. Ms. Crabapple, a teacher at Springfield Elementary, is sponsoring a school trip to the fair. The total cost for transportation, parking and entrance fees is $1200. The total cost will be divided among all the students who go on the trip. a. Complete the table to show how much each student must pay to cover expenses. b. Determine an equation that describes the table of data c. Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in negative values and solve for cost) d. Describe the domain and range. Are there any limiting values of the function? e. How are the equations of questions 1 and 2 similar? Different? f. How are the graphs of questions 1 and 2 similar? Different? Activity taken from © 2008National Council of Teachers of Mathematics http://illuminations.nctm.org The functions in Questions 1 and 2 are known as rational functions. Rational functions are quotients of two polynomials functions. In other words f(x)= p( x) , where p(x) and q(z) are polynomials and q≠0. q( x) Why can q(x) ≠0? TRIP TO THE FAIR PART TWO Ms. Crabapple was just informed that the new attraction, Light It Up will cost an additional $5 per student. a. Complete the table to reflect the change to the amount each student must pay to cover expenses . b. Write an equation that describes the table. c. Some rational functions can be written in the form of y a k , where a,h, and k are constants. If needed, xh rewrite your equation from part b in this form. d. Graph your equation. Then, extend your graph to include negative values in the domain.(you will have to plug in negative values and solve for cost) e. What is your domain and range and how do they relate to the problem? Activity taken from © 2008National Council of Teachers of Mathematics http://illuminations.nctm.org VARIATIONS Rational functions are a type of inverse variation. There are two main types of variations; direct and inverse DIRECT A direct variation is a relationship in which as one variable increases , the other variable increases proportionally or as one variable decreases the other variable decreases proportionally.Basically the two variables ‘react’ in the same manner. A direct variation equation is linear and can be written in the form of y=mx+b where b=0 and the constant of variation is the slope. The graph of a direct variation always passes through the origin(this is what MAKES it proportional). The general form of a direct variation is y=kx k represents the constant of variation INVERSE An inverse variation describes a situation in which one quantity increases as the other decreases proportionally (like the amount of time you are on facebook vs your algebra II grade) . The two variables ‘react’ oppositely. The general form of a direct variation y = k/x where k≠0 k represents the constant of variation At times you may see the term joint variation. This represents a direct relationship among 3 variables where the equation would be y=kxz where k is the constant of cariations COMBINED A combined variation is a relationship that contains both direct and inverse variation. Quantities that vary directly appear in the numerator and quantities that vary inversely are in the denominator EXAMPLES a. Given y varies directly as x, and y = 14 when x = 3.5. Write and graph the direct variation function. y=kx 14=k(3.5) k=14/3.5 k=4 y=4x b.Given y varies inversely as x, and y=2 when x=5. Write and graph the direct variation function. y=k/x 2=k/5 k=6 y=6/x Identify the type of variation and then write the equation. COMPLETE THE GOOGLE SURVEY! LINK IN HAIKU! Activity taken from © 2008National Council of Teachers of Mathematics http://illuminations.nctm.org