We need to determine the image of the points $A(4,1)$ and $B(0, -2)$ after performing the following transformations:

1. 90° counterclockwise rotation about the origin
2. Reflection over the line $y = x$

Step 1: 90° Counterclockwise Rotation About the Origin

The general rule for rotating a point $(x, y)$ 90° counterclockwise about the origin is: $(x, y) \rightarrow (-y, x)$

Apply to Point A(4, 1):

$A(4, 1) \rightarrow A'(-1, 4)$

Apply to Point B(0, -2):

$B(0, -2) \rightarrow B'(2, 0)$

Step 2: Reflection Over the Line $y = x$

The rule for reflecting a point over the line $y = x$ is: $(x, y) \rightarrow (y, x)$

Apply to Point A'(-1, 4):

$A'(-1, 4) \rightarrow A''(4, -1)$

Apply to Point B'(2, 0):

$B'(2, 0) \rightarrow B''(0, 2)$

Final Answer

• The image of point $A(4, 1)$ is $A''(4, -1)$.
• The image of point $B(0, -2)$ is $B''(0, 2)$.

Thus, the new locations of the points after the transformations are $A''(4, -1)$ and $B''(0, 2)$.

Would you like further clarification or more details? Here are 5 questions to extend your understanding:

1. What is the formula for rotating a point by 180° about the origin?
2. How does the reflection change when reflecting over the x-axis instead of the line $y = x$?
3. How do you find the midpoint of a line segment between two points?
4. What is the effect of a 90° clockwise rotation on a point?
5. How can you verify the accuracy of a transformation using matrices?

Tip: When performing multiple transformations, always follow the order of operations step by step to avoid confusion.