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Transformations: Slots

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(Transformations and Trig. Modelling)

(Transformations and Trig. Modelling)

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  • 1. Transformations: Slots
    • Situation: So Leo is playing a slot machine at the casino and it appears that the slot machine sort of works in a pattern. The maximum amount you can receive is 300 coins and the minimum is -100 coins. Let's say that Leo played the slots and the very first thing that happened was that he got -100 coins. After 50 tries (first one counts as a try), Leo gets 300 coins.
  • 2. Question the First
    • a) What would be the cosine graph and the sine graph of the slot machine?
  • 3. Here’s a Little Help
  • 4. Cosine Graph
  • 5. Sine Graph
  • 6. Question the Second
    • b) What would be the equations of said graphs?
  • 7. Remember!
    • The equation of a cosine graph looks like this:
    • f(x) = A cos B (x- C )+ D
    • The equation of a sine graph looks like this:
    • f(x) = A sin B (x- C )+ D
  • 8. Remember!
    • Parameter A : distance of max. and min. values from the sinusoidal axis.
    • Parameter B : NOT the period but determines the period according to this relation:
    • Parameter C : phase shift (horizontal shift). Warning: watch the sign of C.
    • Parameter D : vertical shift (average value). Determines the sinusoidal axis.
    period= 2pi/B OR B=2pi/period A period is the duration of one complete cycle of a wave.
  • 9. Cosine Solution
    • f(x) = -200cos((pi/50)(x))+100
    • If you have a graph it's pretty easy to find the equation. If you don't, you could also solve it in a different way. For the cosine graph, your "A" is -200 because a cosine graph starts at it's max . In this question, it starts at it's min. so it's negative and it's 200 because from the sinusoidal axis it's 200 units above and below (the sinusoidal axis being 100 and your max. is 300 and your min. is -100). It's cos because durrh, it's a cosine graph. It's pi over 50 because to find "B" it’s 2pi over the period (in this case is 100) which reduces to pi/50. You add 100 because the graph is shifted 100 units up because your max. is 300 and your min. is -100. (If you didn’t add the 100, your graph’s max. value would be 200 and it's min. value would be -200.)
  • 10. Sine Solution
    • f(x) = 200sin((pi/50)(x-25))+100
    • In this graph, "A" is going to be 200 because from the sinusoidal axis it's 200 to the max. value and 200 to the min. value. It is positive because a regular sine graph would go from the sinusoidal axis (avg. value) to the max. value, which this graph does. So pi/50 basically has the same reasoning as from the cosine graph (B=2pi/period). This time your "C" is 25 because the graph is shifted 25 units to the right (warning: watch the sign of C! If the graph is shifted to the right, your C becomes negative when inputted into the equation and vice versa.). Your "D" is 100 with the same reasoning from the cosine graph.
  • 11. Question the Third
    • c) What would the equation be if there was a person who played this machine 22 times before Leo?
  • 12. Cosine Solution
    • f(x) = -200cos((pi/50)(x-22))+100
    • So basically what happened here was that you added 22 to your "C" value in the equation in the previous question. You did that because the person before Leo used the slot machine 22 times therefore shifting the graph 22 units to the right. (Remember that your "C" value becomes negative when inputted into the equation.)
  • 13. Sine Solution
    • f(x) = 200sin((pi/50)(x-47))+100
    • The same thing happened here, you added 22 to your "C" value to get 47 (25 +22).