1. The following diagram shows part of the graph of y=p⋅sin(qx)+r. The point A (15, 2) is a maximum point and B(45, 1) is a minimum point. Find the value of p, q, and r.

y = (1/2)*sin(x*pi/30) + 3/2

Step-by-step explanation:

We have the equation:

y = p*sin(q*x) + r.

in this case, p is the amplitude, which is calculated as half of the difference between the value at the peak (the maximum) and the value at the troughs (the minimums).

q is related to the period

r is the mid value of the function, will be equal to the minimum plus the amplitude (or the maximum minus the amplitude).

We know that the maximum is (15, 2) (x = 15, y = 2)

We know that the minimum is (45, 1) (x = 45, y = 1)

Then the value of y at the peak is 2, and the value of y at the trough is 1.

This means that the amplitude is:

p = (2 - 1)/2 = (1/2)

And we know that r is equal to the minimum plus one time the amplitude, then:

r = 1 + 1/2 = 3/2.

Then, for now, our equation is:

y = (1/2)*sin(q*x) + 3/2.

Now we can use the information that  (15, 2) is a maximum.

We know that the maximum value of the function sin(x) is 1.

and it is when:

x = (pi/2) + n*2*pi   (where n is a whole number)

Then we must have that:

15*q = (pi/2) + n*2*pi

as this is periodic, we can define n = 0 and it will be the same.

15*q = pi/2

q = (pi/2)/15 = pi/30.

Now, let's test this with the minimum.

The minimum of the sin(x) function is when:

x = (3/2)*pi + k*2*pi (where k can be any whole number)

and we have a minimum at x = 45, then:

q*45 = (3/2)*pi + k*2*pi

(pi/30)*45 = (3/2)*pi + k*2*pi

(45/30) = (3/2) + k*2

3/2 = 3/2 + k*2

If we take k = 0, then the equality is true

Then q  = pi/30 is consistent.

So we can conclude that the equation is:

y = (1/2)*sin(x*pi/30) + 3/2