The image shows several triangles on a coordinate grid along with two questions related to transformations:

1. (a) Triangle $\text{ABC}$ is mapped into triangle $A_1B_1C_1$ by a reflection. Find the equation of the axis of reflection.

2. (b) Triangle $\text{ABC}$ is mapped onto triangle $A_2B_2C_2$ by a certain single transformation. Describe fully the single transformation.

Analysis and Solution

(a) Finding the Equation of the Axis of Reflection

• The original triangle $\text{ABC}$ has vertices $A(0, 2)$, $B(-2, 0)$, and $C(-1, -1)$.
• The reflected triangle $A_1B_1C_1$ has vertices $A_1(0, -2)$, $B_1(-2, 0)$, and $C_1(-1, 1)$.

The axis of reflection should be the line that is equidistant to corresponding points of $\text{ABC}$ and $A_1B_1C_1$. This axis is the horizontal line through $y = 0$.

Equation of the axis of reflection: $y = 0$

(b) Describing the Single Transformation

• Triangle $\text{ABC}$ is mapped onto triangle $A_2B_2C_2$ by a translation.
• The translation can be found by comparing corresponding points, e.g., $A(0, 2)$ maps to $A_2(0, -4)$.
• This implies a vertical shift of 6 units downwards.

Description of the transformation:

• Translation 6 units downwards.

Would you like further details or have any questions? Here are some related questions you might want to explore:

1. How can I find the equation of the axis of symmetry for other transformations?
2. How is a reflection different from a rotation in terms of properties?
3. Can a transformation involve a combination of reflection and translation?
4. How would you write the matrix representation for a reflection over $y = 0$?
5. What is the difference between a translation and a glide reflection?

Tip: When identifying transformations, always look at how the coordinates change between the original and transformed shapes.