The table shows the height, in feet, of the tide over time, in hours. i know the answer is that it should be cosine instead of sine, but i don't how to determine that. i'd like to know so i could solve questions like these in the future.

For this case we have the following function:
 f (x) = - 4sine ((pi / 6) x) + 5
 We evaluate the function for x = 0
 f (0) = - 4sine ((pi / 6) 0) + 5
 f (0) = -4 * (0) + 5
 f (0) = 5
 We note that the result does not correspond to the table.
 Let's rewrite the function:
 f (x) = - 4cosine ((pi / 6) x) + 5
 We evaluate the function for x = 0
 f (0) = - 4cosine ((pi / 6) 0) + 5
 f (0) = -4 * (1) + 5
 f (0) = 1
 The result corresponds to the table
 
 The student's mistake is to use the sine function instead of using the cosine function.
 The correct function is:
 f (x) = - 4cosine ((pi / 6) x) + 5

The answer is

Step-by-step explanation:

In this case, we have to find the common factor of both terms.

The common factor between the coefficients is 17 because 34=17*2.

Then, the common factor between the variables is "x" because it is in both terms.

Finally the common factor is "17x". With this factor, we have to find the multiplies to get the original terms.

First term:

17x*1=17x

Second term:

17x*(-2y)=-34xy

Finally, the equivalent expression is: