Question

How to do this?

Answer 1

The value of "a" can be found by subtracting the y coordinate of the center from the y coordinate of the peak:

`a = y_"peak"-y_"center"`

From the graph, we observe that `y_"peak" = 9` and `y_"center"=3`

`a = 9-3`

`a = 6`

The value of "b" is found, using the equation,

`bx = 2pi`

and the first value of x where the graph starts to repeat. From the graph, we observe that the graph starts to repeat at the x coordinate `x = pi`

Substitute this value for x into the equation:

`b(pi) = 2pi`

Solve for b:

`b = 2`

The value of "c" is the y coordinate of the center:

`c = y_"center"`

We have already observed that this is 3:

`c = 3`

Substitute the known values into the equation:

`y = 6sin(2x) + 3`

To find the smallest value of x where y = 0 set the above equation equal to 0:

`0 = 6sin(2x) + 3`

Solve for x:

`6sin(2x) = -3`

`sin(2x) = -1/2`

The sine function is negative in the 3rd and 4th quadrants. We want smallest value, therefore, we must make a correction for the 3rd quadrant, when we use the inverse sine function:

`2x = pi - sin^-1(-1/2)`

`x=1/2(pi - sin^-1(-1/2))`

`x = 1/2(pi- (-pi/6))`

`x = 1/2((7pi)/6)`

`x = (7pi)/12`