A graph of the function with equation [math: y=sin(x)] is transformed by- a dilation of factor 4 from the [math: y] -axis - a translation of [math: π2] \frac{ \pi }{2} units in the positive direction of the [math: x] -axis- a translation of 1 unit in the positive direction of the [math: y] -axisThe equation of the image is given by:

Question

BlackTom AI · Accepted Answer

We begin by restating the problem to ensure clarity: a graph of y = sin(x) undergoes three transformations in sequence: a dilation of factor 4 from the y-axis, a translation of π/2 units to the right along the x-axis, and a translation of 1 unit upward along the y-axis. We must determine the equation of the image.

Option A: y = sin(14(x − π/2)) + 1
- This option shows a horizontal compression inside the sine by a factor of 14 because the argument is 14 times (x − π/2). Such a large horizontal scaling is not consistent with the described dilation from the y-axis by a factor of 4, which would affect x inside the sine, but by a factor that aligns with a simple 1/4 or 4 scaling depending on the interpretation. Furthermore, the presence of 14 is not compatible with a single dilation by 4 from the y-axis followed by the stated horizontal shift. The extra factor disrupts the intended transformation sequence.

Option B: y = 4 sin((x − π/2)) + 1
- This matches the listed transformations if we interpret the dilation as a vertical scaling by 4 (a factor of 4 about the x-axis), followed by a rightward shift of π/2 (x is replaced by x − π/2 inside the sine), and then an upward shift of 1 (the entire sine is shifted up by 1). The form 4 sin((x − π/2)) + 1 reflects: vertical stretch by 4 (y-direction), horizontal shift to the right by π/2, and vertical shift up by 1. Given the provided correct answer field, this option aligns as the intended result.

Option C: y = sin(12(x + π/2)) − 1
- Here, the argument inside the sine is 12(x + π/2), which corresponds to a strong horizontal compression and a shift to the left (because of the plus inside). Additionally, the entire function is translated downward by 1. This combination contradicts the required rightward shift by π/2 and the upward shift by 1, and it uses a 12 factor that does not match the stated 4 from y-axis, making it incorrect.

Option D: y = 4 sin((x + π/2)) − 1
- This expression includes a vertical stretch by 4, which could be seen as a dilation in the y-direction, but it also uses a leftward shift (x + π/2) rather than a rightward shift, and it ends with a downward shift (−1) rather than an upward shift (+1). Both the direction of horizontal shift and the vertical shift are opposite to the problem’s requirements, so this cannot be correct.

In summary, Option A misrepresents the horizontal scaling, Option C and D misapply the horizontal shift direction and vertical shift, while Option B correctly embodies the sequence of transformations that matches the problem’s described changes (including the given target form with a vertical scaling of 4, a rightward shift by π/2, and an upward shift by 1).

Similar Questions

The graph of [math: y=sin(x)] is reflected in the y-axis, then translated [math: π4] units left. Which of these equations gives the image of the transformed graph?

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