Hi, I asked this question yesterday and I am still struggling with it, can someone please help me, 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.
I did see that question yesterday, but was waiting until some other math tutor might understand what you mean by "a stretch by a factor of 1/2 about the line y=1 ".
I am going to guess that you mean the distance of the point (2,7) to the line y = 1 cut in half.
After that the new point would be (2,4)
Now this new point is -1 units from x=3, so I would guess we move it 3(-1), which would place it at (-1,4)
I don't know if this is what you ask?
Sure, I can help you with that! To find the image of the point (2,7) after the given transformations, we need to apply each transformation step by step.
Step 1: Vertical stretch by a factor of 1/2 about the line y = 1
To stretch vertically by a factor of 1/2, we need to multiply the y-coordinate of the point by the scaling factor. Since the axis of stretch is y = 1, we need to move the point vertically to the line y = 1 before applying the stretch.
To move the point to the line y = 1, we need to calculate the vertical displacement between the point and the line. In this case, since the point (2,7) is above the line, the vertical displacement is (7 - 1) = 6 units downward.
Now, subtracting the vertical displacement from the y-coordinate, we get y' = 1 - 6 = -5. So, after moving the point to the line y = 1, the new coordinates become (2, -5).
Next, we apply the vertical stretch by multiplying the y-coordinate by the factor of 1/2. So, y'' = (-5) * (1/2) = -2.5.
After the vertical stretch, the new coordinates become (2, -2.5).
Step 2: Horizontal stretch by a factor of 3 about the line x = 3
To stretch horizontally by a factor of 3, we need to multiply the x-coordinate of the point by the scaling factor. Since the axis of stretch is x = 3, we need to move the point horizontally to the line x = 3 before applying the stretch.
To move the point to the line x = 3, we need to calculate the horizontal displacement between the point and the line. In this case, since the point (2, -2.5) is to the left of the line, the horizontal displacement is (3 - 2) = 1 unit to the right.
Now, adding the horizontal displacement to the x-coordinate, we get x' = 3 + 1 = 4. So, after moving the point to the line x = 3, the new coordinates become (4, -2.5).
Next, we apply the horizontal stretch by multiplying the x-coordinate by the factor of 3. So, x'' = (4) * (3) = 12.
After the horizontal stretch, the final image of the point (2,7) is (12, -2.5).
I hope this explanation helps you understand how to find the image of a point after a series of transformations! Let me know if you have any further questions.