We want to find a sine function such that:
The given options are:
Notice that the first information that we have, implies that y must be equal to 2 when x = 0.
Knowing that we can discard the third option which has a y-intercept of 4.
Now the second. Remember that for a general function:
y = A*sin(k*x) + M
Where A is the amplitude and M is the midline, the maximum and minimum are given by:
A + M
-A + M
In all the remaining options we have:
A = ± 3
M = 2
We can see that with these we get the maximum and minimum of 5 and -1.
Finally, we know that it goes through one cycle at pi.
This means that if f(x) is the function, then:
f(x) = f(x + pi)
Also remember that for a general sine function, we have:
sin(x) = sin(x + 2*pi)
Ok, now let's analyze the options (only the sine part)
1) sin(2*(x + pi)) = sin( 2*x + 2*pi) = sin(2*x)
if we evaluate this in x + pi, we will get:
sin(2*( x + pi + pi)) = sin(2*(x + 2*pi)) = sin( 2*x + 4*pi) = sin(2*x)
So yes, option 1 is a correct option (you also can see that option 4 has the exact same sine part, so that option is also correct.)
2) for the second option the sine part is:
sin(x + pi)
if we evaluate this in x + pi we get:
sin(x + pi + pi) = sin(x + 2*pi) = sin(x)
and we have:
sin(x +pi) ≠ sin(x)
Then this function does not go through one cycle at pi.
We can conclude that the two options that meet all the conditions are the first one and fourth one:
1) y = -3*sin(2*(x + pi)) + 2
4) y = 3*sin(2*(x + pi)) + 2
We can't say which one is correct if we do not look at the graph.
If the function starts increasing, then (2) is the correct one, if the graph starts decreasing, then (1) is the correct one.
Below, you can see a graph of both functions:
The green one is y = 3*sin(2*(x + pi)) + 2, the blue one is y = -3*sin(2*(x + pi)) + 2
If you want to learn more, you can read:
https://brainly.com/question/14068845
