Question

What is the general equation of a sine function with an amplitude of 6, a period of π/4 and a horizontal shift of π/2?

Answer 1

The formula for the general sin function is $y=Asin(bx+c)$ where b is the period and c is the phase shift.  We are given a period of $\frac{ \pi }{4}$ and the formula to solve for b is $period= \frac{2 \pi }{b}$.  We set up to solve for b: $\frac{ \pi }{4} = \frac{2 \pi }{b}$.  Cross multiply to get $b \pi =8 \pi$ and b = 8.  Now for the phase shift. The formula for that is -c/b, and since we already know b, and we have a phase shift value of pi/2, we solve for c: $\frac{ \pi }{2}= \frac{-c}{8}$ and $8 \pi =-2c$.  Therefore, $c=-4 \pi$.  Putting all that together into the equation with the A value of 6, we have $y=6sin(8x-4\pi)$.  Factoring out the 8 then, the final equation is $y=6sin(8(x- \frac{\pi}{2} ))$, third choice down.

Answer 2

The general equation of a sine function with an amplitude of 6, a period of π/4, and a horizontal shift of π/2 is $\rm y = 6\;sin(8x-4\pi)$.

Given :

Sine function with an amplitude of 6, a period of π/4, and a horizontal shift of π/2.

The following steps can be used in order to determine the general equation of a sine function:

Step 1 - The generalized equation of sine function is given below:

$\rm y = Asin(cx+d)$

where A is the amplitude, c is the time period and d is the horizontal shift.

Step 2 - According to the given data, the amplitude of 6, a period of π/4, and a horizontal shift of π/2.

Step 3 - The value of b is calculated as:

$\rm \dfrac{\pi}{4}=\dfrac{2\pi}{b}$

b = 8

Step 4 - The value of 'c' is calculated as:

$\rm 8\pi = -2c$

$c = -4\pi$

Step 5 - Substitute the value of the known terms in the above expression.

$\rm y = 6\;sin(8x-4\pi)$

Therefore, the correct option is C).

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