https://file.aiquickdraw.com/mathbot/file/1727351399825qw20lbeb.jpg

To find the gradient (slope) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \), we use the formula:

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

Let's solve for each pair of points in the image:

1. **For \( A(0, -4) \) and \( B(4, 4) \):**
   \[
   \text{Gradient} = \frac{4 - (-4)}{4 - 0} = \frac{4 + 4}{4} = \frac{8}{4} = 2
   \]

2. **For \( C(0, 0) \) and \( D(2, -4) \):**
   \[
   \text{Gradient} = \frac{-4 - 0}{2 - 0} = \frac{-4}{2} = -2
   \]

3. **For \( M(-4, 0) \) and \( N(-2, 4) \):**
   \[
   \text{Gradient} = \frac{4 - 0}{-2 - (-4)} = \frac{4 - 0}{-2 + 4} = \frac{4}{2} = 2
   \]

4. **For \( X(-6, -2) \) and \( Y(-3, -5) \):**
   \[
   \text{Gradient} = \frac{-5 - (-2)}{-3 - (-6)} = \frac{-5 + 2}{-3 + 6} = \frac{-3}{3} = -1
   \]

### Final gradients:
1. \( A \) and \( B \): 2
2. \( C \) and \( D \): -2
3. \( M \) and \( N \): 2
4. \( X \) and \( Y \): -1

Would you like more details on any of these calculations?

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**Follow-up questions:**
1. What is the meaning of a positive gradient?
2. What happens if the gradient is zero?
3. How would you find the equation of a line given two points?
4. Can two different pairs of points have the same gradient?
5. What is the relationship between gradients and parallel lines?

**Tip:** When calculating gradients, always subtract coordinates consistently to avoid mistakes.

Find the gradient of the line between each pair of points: A(0, -4) and B(4, 4); C(0, 0) and D(2, -4); M(-4, 0) and N(-2, 4); X(-6, -2) and Y(-3, -5).

Learn how to find the gradient of a line between pairs of points such as A(0, -4) and B(4, 4). This tutorial covers important concepts like the slope formula and offers detailed step-by-step solutions for each pair of points. Suitable for Grades 7-9.

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