Question

Find the equation of a sine wave that is obtained by shifting the graph of y = sin(x) to the right 3 units and dowriward 3 units and is vertically stretched by a factor of 8 when compared to y = sin(x) y = __

Answer 1

The equation of the transformed sine wave with a horizontal shift to the right by 3 units, a vertical stretch by a factor of 8, and a vertical shift downward by 3 units is y = 8sin(x - 3) - 3.

Step-by-step explanation:

The student has been asked to find the equation of a sine wave that has been modified from its original form y = sin(x). The modifications include a horizontal shift to the right by 3 units, a vertical shift downward by 3 units, and a vertical stretch by a factor of 8. The new equation reflecting these changes is given by:

y = 8sin(x - 3) - 3

To obtain this equation, we recognize that the amplitude is multiplied by 8, the horizontal shift of 3 units to the right is represented by x minus 3 in the argument of the sine function, and the vertical shift of 3 units downward is represented by subtracting 3 from the entire function.

Answer 2

To shift the graph of y = sin(x) to the right 3 units, downward 3 units, and vertically stretch it by a factor of 8, the equation becomes y = 8sin(x-3) - 24.

Step-by-step explanation:

The equation for the desired sine wave can be constructed by applying the transformations to the original equation y = sin(x) in the following order:

Vertical stretch: Multiplying the original equation by a factor of 8 stretches the wave vertically by the same factor. So, we begin with:

y = 8 × sin(x)

Horizontal shift: Shifting the graph 3 units to the right is equivalent to replacing x with x - 3 in the equation. So, we have:

y = 8 × sin(x - 3)

Vertical shift: Shifting the graph 3 units downward is equivalent to adding -3 to the entire equation. Therefore, the final equation becomes:

y = 8 × sin(x-3) - 24

This equation represents a sine wave with an amplitude of 8, shifted 3 units to the right and 3 units downward relative to the original y = sin(x) graph.