https://file.aiquickdraw.com/mathbot/file/1725564729971bsyxnh6l.pdf

Here are the questions and instructions from the document, along with my answers to the six questions:

1. **In which quadrant does the point (-4, -3) lie?**  
   - **Answer:** The point (-4, -3) lies in the **third quadrant**, as both the x and y coordinates are negative.

2. **Choose the correct word to complete the statement:**
   - **If the gradients of two straight lines are equal, the lines are (parallel/perpendicular).**
   - **Answer:** If the gradients of two straight lines are equal, the lines are **parallel**.

3. **If the gradient of a line is negative, then the line is said to be (increasing/decreasing).**
   - **Answer:** If the gradient of a line is negative, the line is said to be **decreasing**.

4. **Write down the equation of a vertical line that cuts the x-axis at 2.**
   - **Answer:** The equation of a vertical line that cuts the x-axis at 2 is **x = 2**.

5. **Calculate the gradient between the points (1, -1) and (3, 5).**
   - **Answer:** The formula for the gradient (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

     \[
     m = \frac{y_2 - y_1}{x_2 - x_1}
     \]

     Substituting the values:

     \[
     m = \frac{5 - (-1)}{3 - 1} = \frac{6}{2} = 3
     \]

     So, the gradient is **3**.

6. **Write down the equation of the line through these points if the y-intercept is -4.**
   - **Answer:** Using the slope-intercept form \( y = mx + c \), where \( m \) is the gradient and \( c \) is the y-intercept:
   
     \[
     y = 3x - 4
     \]

Would you like more details or explanations on any of these? 

Here are five related questions to further expand your knowledge:

1. How do you determine the gradient of a line given two points?
2. What is the significance of the sign of a gradient in graphing?
3. Can two perpendicular lines have the same gradient? Why or why not?
4. How can you determine the equation of a line if given a point and the gradient?
5. What is the relationship between the gradients of parallel and perpendicular lines?

**Tip:** Remember, the gradient of a line defines how steep or flat the line is.

Learn how to solve problems involving gradients and equations of lines, including determining quadrants, calculating gradients, and writing equations. This example covers key concepts for Grade 9 students.

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