Which of the following sequences of transformations maps
figure 1 to figure 2?
90° clockwise rotation around
the origin, then a reflection
across the x-axis
90° counterclockwise rotation
around the origin, then a
reflection across the x-axis
-4
reflection across the y-axis, then
a reflection across the x-axis
reflection across the x-axis, then
a reflection across the y-axis

First choice: 90° clockwise rotation around the origin, then a reflection across the x-axis.

Explanation:

The vertices of the figure 1 are:

(-3, 2)(-4, 5)(-3, 8)(-7,5)

The vertices of figure 2 are:

(2, -3)(5, -4)(8, -3)(5, -7)

By simple inspection, you can tell that if you rotate the figure 1 90º clockwise and then reflect the image across the x-axys you obtain the figure 2.

You can prove that analitically.

1. Rotation 90º clockwise

A 90º clockwise reotation is the same that a 270º counterclockwise rotation and the mathematical rule for this is:

(x,y) → (y, x)

Applying that rule to the vertices of figure 1, you obtain:

(-3, 2) → (2, 3)(-4, 5) → (5, 4)(-3, 8) → (8, 3)(-7,5) → (5, 7)

2. Reflection across the x-axis.

A reflection across the x-axis keeps the same x-coordinate and changes the sign of the y-coordinate. The rule is:

(x,y) → (x, -y)

Applying that rule to the previous points yields to:

(2, 3) → (2, -3)(5, 4) → (5, -4)(8, 3) → (8, -3)(5, 7) → (5, -7)

Which are the coordinates of the figure 2. Thus, you have proved that the sequence of transformations 90° clockwise rotation around the origin, then a reflection across the x-axis (first choice) maps the figure 1 to figure 2.

1. The given rectangular equation is .

We substitute .

Divide through by

2. The given rectangular equation is:

This is the same as:

We use the relation

This implies that:

3. The given rectangular equation is:

This is the same as:

We use the relation and

This implies that:

Divide through by r

4. We have

We substitute and

This implies that;

5. We have

We substitute and

This implies that;