Let’s solve the questions step by step.

Question 6:

Which of the following is the equation for a line that is parallel to the $x$-axis that passes through the point $(5, -4)$?

Solution:

A line parallel to the $x$-axis has a constant $y$-coordinate because it does not move up or down. The general equation is: $y = c$ where $c$ is the $y$-coordinate of any point on the line. Since the line passes through the point $(5, -4)$, the $y$-coordinate is $-4$.
Thus, the equation is: $y = -4$

Answer:

Option (3) $y = -4$

Question 7:

Which of the following is the image of the coordinate $(-3, -5)$ mapped under the rule $(x, y) \to (x+8, y-2)$?

Solution:

The transformation rule $(x, y) \to (x+8, y-2)$ means:

1. Add $8$ to the $x$-coordinate.
2. Subtract $2$ from the $y$-coordinate.

Starting with the point $(-3, -5)$:

• $x' = -3 + 8 = 5$
• $y' = -5 - 2 = -7$

Thus, the new coordinate is: $(5, -7)$

Answer:

Option (2) $(5, -7)$

Question 8:

If the coordinate $B(6, -3)$ was rotated $90^\circ$ clockwise about the origin, its image would be located at which of the following?

Solution:

To rotate a point $90^\circ$ clockwise about the origin, the transformation rule is:
$(x, y) \to (y, -x)$ Here, the original point is $B(6, -3)$:

• $x = 6$, $y = -3$
  Using the rule:
• $x' = y = -3$
• $y' = -x = -6$

Thus, the new coordinate is: $(-3, -6)$

Answer:

Option (1) B'(-3, -6)

Summary of Answers:

1. Question 6: $y = -4$ (Option 3)
2. Question 7: $(5, -7)$ (Option 2)
3. Question 8: $B'(-3, -6)$ (Option 1)

Do you need further clarification or detailed explanations on any step?

Related Questions:

1. What is the general equation of a vertical line?
2. How do you determine the equation of a line parallel to the $y$-axis?
3. What is the effect of rotating a point $180^\circ$ clockwise about the origin?
4. How can you derive a transformation rule for reflecting a point across the $x$-axis?
5. What are the rules for translating a point in the coordinate plane?

Tip:
For any line parallel to the $x$-axis, the $y$-coordinate remains constant across all points.