Question

Write an equation with amplitude 2, period (T) pi/2, phase shift (ø) - pi/4, and a midline y=-3.

Im mostly struggling with the period.

Answer 1

The amplitude is 2, so A = 2 (see note 1 below)

The period is T = pi/2 which means
B = 2pi/T
B = 2pi/(pi/2)
B = 2pi*(2/pi)
B = 4

The phase shift is C = pi/4
The midline is y = -3 indicating that D = -3

Put this all together and we go from this (see note 2)
y = A*sin(B(x-C)) + D
to this
y = 2*sin(4(x-pi/4)) + (-3)
and that simplifies to
y = 2sin(4x-pi) - 3
which is one way to write the final simplified answer (see note 3)

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Extra Stuff:
Note 1: The value of A can be A = -2. Recall that the amplitude is equal to |A|. So If A = -2, then |A| = |-2| = 2. The amplitude is the vertical distance from the midline to the peak or valley. Distance can't be negative. To make things simple, I went with A = 2.

Note 2: The general formula involving sine can be replaced with cosine instead. The cosine function is basically sine but a phase shift of it (it has the same shape and pattern, but has been moved over). You cannot use tangent as that is a completely different class of function.

Note 3: I distributed the 4 through to the (x-pi/4) to get 4x-pi. Your teacher or book may want you to keep things factored and not distribute. If so, then the answer would be y = 2sin(4(x-pi/4)) - 3. That is of course you went with using sine instead of cosine. The benefit of not distributing is that we can more easily see what the phase shift is.

Answer 2

Sure thing! Here's how you can write this equation:

First, let's look at the general form of a sine function:

y = A * sin(B*(x - ø)) + D

Where:
- A is the amplitude
- B = 2pi / T
- ø is the phase shift
- D is the midline

The amplitude tells us the peak deviation of the function from its midline, so here it will be 2.

The period is pi/2, so that means B, which equals 2pi divided by the period, will be 2pi/(pi/2) = 4.

The phase shift is negative pi/4. This value gets subtracted from x inside the sine function, effectively shifting the graph to the right by pi/4.

The midline of the graph is -3. This vertical shift is represented by D, which is -3.

Now insert these into the general form:

y = 2 * sin(4*(x - (-pi/4))) - 3

Or simplifying:

y = 2 * sin(4x + pi) - 3

This equation represents a sine curve with an amplitude of 2, a period of pi/2, shifted to the right by pi/4, and shifted downward by 3 units.