PreCalc

Hi, I'm needing help on how to find the vertical shift, horizontal shift, amplitude, period, domain, and range; As well as the sine and cosine function based off of 12 data points (Which i'll provide below) Can someone please walk me through the steps on how to find each. Or some hints to find the information please? Thank you.

Data points: (1, 53) (2,58) (3,65) (4, 73) (5,84) (6,92) (7,97) (8,94) (9,90) (10,76) (11,66) (12,52)
*Assume that the data is periodic and use the graph to determine the values mentioned above for both a sine function and cosine function*

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  1. I plotted your points and it looked like this
    https://www.wolframalpha.com/input/?i=Plot+%28%281%2C+53%29+%282%2C58%29+%283%2C65%29+%284%2C+73%29+%285%2C84%29+%286%2C92%29+%287%2C97%29+%288%2C94%29+%289%2C90%29+%2810%2C76%29+%2811%2C66%29+%2812%2C52%29%29

    Since you wanted a trig function, we could treat (1,53) as a minimum and (7,97) as a maximum
    which would give us an amplitude of 97- 53 or 44 and a vertical shift of 75
    the period would be 12
    then 2π/k = 12 ---> k = π/6

    so we could begin with
    y = 22 sin ((π/6)(x + d)) + 75, where d is a horizontal shift

    sub in one of the points, say (5,84)
    84 = 22 sin ((π/6)(5 + d)) + 75
    9 = 22 sin ((π/6)(5 + d))
    .4090909... = sin ((π/6)(5 + d)) , make sure your calculator is set to radians
    (π/6)(5 + d) = .42145...
    5+d = .8049...
    d = appr -4.195

    y = 22 sin (π/6)(x - 4.195) + 75

    testing for one of the other points , say (4,73) to see how close we come
    LS = 73
    RS = 22 sin(π/6(4-4.195) + 75
    = 72.75 , hey, that's not bad

    let's try another: (11,66)
    LS = 66
    RS = 22 sin(π/6(11-4.195) + 75
    = 66, well! how about that, I think we nailed it

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    Reiny
  2. My wording is a bit off in "give us an amplitude of 97- 53 or 44"

    I should have said a range of 44, which would make the amplitude 22
    (no change in the solution, I used 22 for the amplitude)

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    Reiny
  3. Thank you so much Reiny!

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