Let's go through the questions one by one.

Question 1: Write the rule for these graphs.

This question asks for the rules (equations) for four different graphs.

Graph (a):

• The graph shows a straight line passing through the origin (0, 0) and sloping downward. This is a straight line, and the slope seems to be negative.
• The equation should be of the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.
• Since the line passes through the origin, $b = 0$, and the slope appears to be $-1$, meaning the rule for this graph is: $y = -x$

Graph (b):

• This graph shows a straight line with a positive slope passing through the origin (0, 0).
• The slope looks to be positive and the same steepness as in part (a), suggesting the slope is 1.
• The rule is: $y = x$

Graph (c):

• This graph is a horizontal line passing through $y = 2$ on the y-axis.
• A horizontal line has no slope, so the equation is just the y-value.
• The rule is: $y = 2$

Graph (d):

• This graph is a vertical line passing through $x = -3$ on the x-axis.
• A vertical line has an undefined slope, and the equation is based on the x-value.
• The rule is: $x = -3$

Question 2: Sketch the following graphs.

Part (a): $y = -1.5$

• This is a horizontal line where $y$ is always $-1.5$.
• The graph will be a horizontal line crossing the y-axis at $y = -1.5$.

Part (b): $x = 0$

• This is a vertical line where $x$ is always $0$, which is the y-axis.
• The graph will be a vertical line along the y-axis.

Would you like more details or have any questions on these answers?

Here are 5 related questions you could explore:

1. How can you determine the slope of a line from a graph?
2. What is the general equation of a line, and how do you find the slope?
3. How would the graph change if the slope in part (a) of Question 1 was steeper?
4. What is the significance of horizontal and vertical lines in geometry?
5. How do you interpret the intercepts of a line in relation to its graph?

Tip: For vertical lines, the equation will always be $x = \text{{constant}}$, while for horizontal lines, it will always be $y = \text{{constant}}$.