Find the vertical/horizontal shift, amplitude, period, domain, and range of the given values. Find the sine/cosine functions.
(1,67) (2,71) (3,79) (4,86) (5,95) (6,105) (7,106) (8,104) (9,100) (11,76) (12,65)
Initial attempt:
https://www.wolframalpha.com/input/?i=plot+%7B%281%2C67%29+%282%2C71%29+%283%2C79%29+%284%2C86%29+%285%2C95%29+%286%2C105%29+%287%2C106%29+%288%2C104%29+%289%2C100%29+%2811%2C76%29+%2812%2C65%29%7D
"vertical/horizontal shift, amplitude, period, domain" are terms associated with trig functions, so it looks like we have cose to 1 period of a sinusoidal curve
let' go with a sine curve.
the period appears to be 12-1 = 11
so 2π/k = 11
k = 2π/11
the largest I see is 106 and the lowest is appr 65
so a = (106-65)/2 = 20.5
shape of curve = 20.5 sin(2π/11 x)
the max of that is 20.5, but it should be 106 so we have to raise our curve by 85.5
so far: y = 20.5 sin (2π/11 x) + 85.5
Now all we have to move it horizontally left or right for the points to match reasonably well
let y = 20.5 sin (2π/11(x + d)) + 85.5 be such that the point (4,86) will almost lie on it.
86 = 20.5 sin (2π/11(4 + d)) + 85.5
sin (2π/11(4 + d)) = .5/20.5 = .02439
Using my calculator, I found that sin(.02439) = .02439, so
2π/11(4 + d) = .02439
4+d = .04270..
d = -3.9573
how about y = 20.5 sin (2π/11(x -3.9573)) + 85.5
mhhh:
https://www.wolframalpha.com/input/?i=plot++y+%3D+20.5+sin+%282%CF%80%2F11%28x+-3.9573%29%29+%2B+85.5+from+1+to+12
let's test one of the points: (8,104) , hope to get y = 104 , when x = 8
y = 20.5sin(2π/11(8 -3.9573)) + 85.5
= 20.5sin(2π/11(4.0427...)) + 85.5
= 20.5sin(2.30918..)) + 85.5
= 15.16... + 85.5
= 100.66 , should be 104
(that's about a 3% error, what do you think?)