The vertices of a pre-image are located at points (12, 8), (6, -4), and (-2, 10). If the image has coordinates (6, 4), (3, -2), and (-1, 5), is this an expansion or a contraction?
If you notice that all the coordinates of the image are half the corresponding values in the pre-image, this means that the transformation is a contraction about the origin, with a factor of 0.5.
Im sorry, I have no clue what you said. Could you break it down?
It gets smaller.
so Its a contraction?
okay, nvm i got it
ok, always sketch a graph
To determine whether the given transformation is an expansion or a contraction, we need to examine the change in distances between corresponding points in the pre-image and the image.
Step 1: Find the distances between vertices of the pre-image and the image.
- Distance between (12, 8) and (6, 4):
d₁ = √((6 - 12)² + (4 - 8)²) = √((-6)² + (-4)²) = √(36 + 16) = √52 ≈ 7.2
- Distance between (6, -4) and (3, -2):
d₂ = √((3 - 6)² + (-2 - (-4))²) = √((-3)² + 2²) = √(9 + 4) = √13 ≈ 3.6
- Distance between (-2, 10) and (-1, 5):
d₃ = √((-1 - (-2))² + (5 - 10)²) = √(1² + (-5)²) = √(1 + 25) = √26 ≈ 5.1
Step 2: Calculate the ratio between the distances of corresponding points.
- Ratio of d₁ to d₂: d₁/d₂ = 7.2/3.6 = 2
- Ratio of d1 to d₃: d₁/d₃ = 7.2/5.1 ≈ 1.41
Step 3: Analyze the ratios:
- If the ratio is greater than 1, it indicates an expansion.
- If the ratio is less than 1, it suggests a contraction.
In this case:
- The ratio of d₁ to d₂ is 2, which is greater than 1. Therefore, we have expansion in the x and y directions.
- The ratio of d₁ to d₃ is approximately 1.41, greater than 1. Thus, we have expansion in the x and y directions.
Conclusion:
The given transformation is an expansion.