To find the sequence of transformations that maps triangle \( \Delta ABC \) to triangle \( \Delta A"B"C" \), we need to analyze the two triangles' coordinates:
- \( A (-5, 2) \), \( B (-2, 4) \), \( C (-3, 0) \)
- \( A" (-1, -2) \), \( B" (2, -4) \), \( C" (1, 0) \)
Step 1: Translation
To see the translation, we notice that the x-coordinates of \( A \), \( B \), and \( C \) all increase by 4 units.
- From \( -5 \) to \( -1 \): \( -5 + 4 = -1 \)
- From \( -2 \) to \( 2 \): \( -2 + 4 = 2 \)
- From \( -3 \) to \( 1 \): \( -3 + 4 = 1 \)
So, there is a horizontal translation of 4 units to the right.
Step 2: Reflection
Next, we notice the changes in the y-coordinates:
- The y-coordinate at point \( A \) goes from \( +2 \) to \( -2 \) (which indicates reflection and shifting downward).
- The y-coordinate at point \( B \) goes from \( +4 \) to \( -4 \).
- The y-coordinate at point \( C \) goes from \( 0 \) to \( 0 \) (no change).
The reflection indicates flipping about the x-axis, which changes \( y \) to \( -y \).
Step 3: Vertical Translation
After reflecting, we need to shift downwards to align \( y \) values:
- From 2 to -2: After reflecting, we get \( -2 \), and it needs to go down another 4 units.
- From 4 to -4: After reflecting, we have \( -4 \) already.
- From 0 to 0: No further shift is needed.
This means we also translate downward by 4 units.
Final Transformation Composition
Combining these transformations, we first reflect over the x-axis and then translate right by 4 and down by 4. The complete transformation can be represented in function notation as follows:
- Reflect over the x-axis: \( (x, y) \to (x, -y) \)
- Translate right by 4: \( (x, -y) \to (x + 4, -y) \)
- Translate down by 4: \( (x + 4, -y) \to (x + 4, -y - 4) \)
Therefore, the complete transformation can be expressed as: \[ (x, y) \rightarrow (x + 4, -y - 4) \]
Looking at the options, the correct transformation sequence is:
(x,y)→(x,−y−4)
1 answer