Question

Can someone please help me with this question. Thank you!

State the image of the given point after the following transformation:
(2,7) after a vertical stretch by a factor of 1/2 about the line y=1 followed by a horizontal stretch by a factor of 3 about the line x=3.

Answer 1

[AI-generated answer — Explain Bot]

Sure! To find the image of a point after a sequence of transformations, we need to apply each transformation one by one in the given order.

First, let's understand what a vertical stretch by a factor of 1/2 about the line y=1 means. This transformation scales the y-coordinates of the point while keeping the x-coordinates the same. In this case, the line y=1 serves as the axis of the stretch. To perform this stretch, we need to multiply the y-coordinate of the given point by the stretch factor (1/2) while keeping the x-coordinate unchanged.

So, after the vertical stretch, the coordinates of the point (2,7) become (2, 7 * (1/2)) = (2, 3.5).

Next, we apply a horizontal stretch by a factor of 3 about the line x=3. This transformation scales the x-coordinates of the point while keeping the y-coordinates the same. The line x=3 serves as the axis of the stretch. To perform this stretch, we need to multiply the x-coordinate of the point by the stretch factor (3) while keeping the y-coordinate unchanged.

Hence, after the horizontal stretch, the coordinates of the point (2, 3.5) become ((2 * 3), 3.5) = (6, 3.5).

Therefore, the image of the given point (2,7) after the specified transformations is (6, 3.5).

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the image of the point (2, 7) after the described transformations, we can follow these steps:

1. Vertical stretch by a factor of 1/2 about the line y = 1:
Since the line of reflection is y = 1, the distance between the point (2, 7) and the line y = 1 is 7 - 1 = 6 units.
Applying the vertical stretch by a factor of 1/2, the new distance becomes 6 * 1/2 = 3 units.
Since the point (2, 7) is above the line y = 1, the new image will be 3 units below the line y = 1.
Therefore, the new image after the vertical stretch is (2, 1 - 3) = (2, -2).

2. Horizontal stretch by a factor of 3 about the line x = 3:
Since the line of reflection is x = 3, the distance between the point (2, -2) and the line x = 3 is 2 - 3 = -1 unit.
Applying the horizontal stretch by a factor of 3, the new distance becomes -1 * 3 = -3 units.
Since the point (2, -2) is to the left of the line x = 3, the new image will be 3 units to the left of the line x = 3.
Therefore, the new image after the horizontal stretch is (2 - 3, -2) = (-1, -2).

The image of the point (2, 7) after the described transformations is (-1, -2).