Question

Consider the function f(x)= sin(x) which has a period of 2Pi. What modification to the function rule would cause the period to become 8Pi?

Answer 1

The correct modification to have a period of 8Π is division of the x variable by 4 inside the sine function, making the correct option (d).

Step-by-step explanation:

To modify the function f(x) = sin(x) to have a period of 8Π, we would need to look at how the period of a sine function is determined. The general form of a sine function is f(x) = sin(Bx), where 2Π/|B| gives us the period of the sine wave. Since the period we desire is 8Π, we need to find the value of B that makes this true.

The original period is 2Π, so to make the period 8Π, which is four times longer, we need to divide the coefficient of x inside the sine function by 4. Therefore, the modified function will be f(x) = sin(x/4) or f(x) = sin(0.25x). This alteration to the function changes its period, without affecting its range or the fact that it oscillates between +1 and -1.

In summary:

• Original function: f(x) = sin(x), Period: 2Π
• Modified function: f(x) = sin(x/4), Period: 8Π

The correct modification to have a period of 8Π is division of the x variable by 4 inside the sine function, making the correct option (d).

Answer 2

The period of the function f(x) = sin(x) is 2π. To modify the function rule so that the period becomes 8π, you can achieve this by stretching the graph horizontally.

The general form for a horizontal stretch or compression of the sine function is f(x) = sin(bx), where b is a positive constant that affects the period of the function.

The period of the function f(x) = sin(bx) is given by the formula P = (2π)/|b|, where P is the period.

To modify the function rule so that the period becomes 8π, we need to find the value of b such that the period is 8π. We can rearrange the period formula to solve for b:

P = (2π)/|b|
8π = (2π)/|b|

Solving for |b|, we get:
|b| = (2π)/(8π)
|b| = 1/4

So, in order to modify the function rule to achieve a period of 8π, the new function rule would be f(x) = sin((1/4)x). This modification stretches the graph horizontally by a factor of 1/4, effectively increasing the period to 8π.