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*
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
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)
Thank you so much Reiny!
To find the vertical shift, horizontal shift, amplitude, period, domain, and range of a sine or cosine function given data points, we can follow these steps:
1. Plot the given data points on a graph. Label the x-axis as the independent variable (usually time or angle) and the y-axis as the dependent variable (usually amplitude or height).
2. Identify the vertical shift (also known as the phase shift). This refers to any upward or downward shift of the graph. To find the vertical shift, we need to examine the y-coordinates of the data points. In the given data points, find the average of the y-values by adding them up and dividing by the number of data points.
3. Determine the amplitude. The amplitude is the distance from the center line (vertical shift) to the peak or trough of the curve. It can be found by calculating half the range (the difference between the maximum and minimum y-values).
4. Calculate the period. The period is the horizontal length of one complete cycle of the graph. To find this, identify the difference between two consecutive x-values where the function repeats. In your data points, this is 1.
5. Determine the horizontal shift. The horizontal shift is the amount the graph is shifted left or right. In the given data, the minimum x-value is 1. We subtract 1 from each x-value to make it start from zero.
6. Find the domain and range. The domain represents the set of all possible x-values, which is determined by the given data points. The range represents the set of all possible y-values. In the given data points, the domain is the set of x-values, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, and the range is the set of y-values, {52, 53, 58, 65, 73, 76, 84, 90, 92, 94, 97}.
7. Determine whether the given data corresponds to a sine or cosine function. We can infer this by examining the shape of the graph. If the graph starts at a maximum (peak), then it corresponds to a sine function. If the graph starts at a minimum (trough), then it corresponds to a cosine function.
With the given 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)
The average of the y-values is (53+58+65+73+84+92+97+94+90+76+66+52) / 12 = 76.75. This is the vertical shift or phase shift.
The amplitude can be found by calculating half the range, which is (97 - 52) / 2 = 22.5.
The period is 1, as there is a repeat in the x-values after every 1 unit.
The horizontal shift is 1, as we subtracted 1 from each x-value.
The domain is {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, and the range is {52, 53, 58, 65, 73, 76, 84, 90, 92, 94, 97}.
The graph starts at a minimum (trough), so it corresponds to a cosine function.
Therefore, based on the given data points, the cosine function (with the corresponding vertical shift, horizontal shift, amplitude, period, domain, and range) can be written as:
f(x) = 22.5 * cos((x - 1) * (2π/1)) + 76.75