Question

Which function has been vertically stretched by a factor of 6 and horizontally stretched by a factor of 2 from a starting function of g(x) = sin(x)?

a) f(x) = 6sin(2x)
b) f(x) = 6sin(1/2x)
c) f(x) = sin(6x)
d) f(x) = sin(2x)

Answer 1

The function that has been vertically stretched by a factor of 6 and horizontally stretched by a factor of 2 from g(x) = sin(x) is f(x) = 6sin((1/2)x), which is option b).

Step-by-step explanation:

To determine which function has been vertically stretched by a factor of 6 and horizontally stretched by a factor of 2 from the starting function g(x) = sin(x), we need to understand the effect that each transformation has on the function.

Vertical stretching by a factor of 6 multiplies the output values of the function by 6, modifying the amplitude but not the period or phase of the sine wave. This can be represented as 6sin(x).

Horizontal stretching by a factor of 2 is a bit more complex. It involves stretching or compressing the graph along the x-axis. In the case of the sine function, this is achieved by dividing the input to the sine function by the stretching factor. This means that the function sin(x) is transformed to sin(x/2) for a horizontal stretch by a factor of 2.

Combining both transformations, we get a function that has been vertically stretched by a factor of 6 and horizontally stretched by a factor of 2 from g(x) = sin(x). The correct function is therefore f(x) = 6sin(​​(1/2)x), which corresponds to option b).