Question

Find the gradient of the following and hence represent it graphically using the following values:
1. A (2,5) B(-2,-5)
2. P (2,9) Q(-5,-5)
3. D (1,3) E(4,-3)

Answer 1

To find the gradients of the lines formed by the given points, we use the formula for the gradient (or slope) \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

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

Now, let's calculate the gradient for each pair of points.

### 1. Points A (2,5) and B (-2,-5)

Let \( A(2,5) \) and \( B(-2,-5) \):
- \( x_1 = 2 \), \( y_1 = 5 \)
- \( x_2 = -2 \), \( y_2 = -5 \)

Calculating the gradient \( m \):

\[
m = \frac{-5 - 5}{-2 - 2} = \frac{-10}{-4} = \frac{10}{4} = \frac{5}{2}
\]

### 2. Points P (2,9) and Q (-5,-5)

Let \( P(2,9) \) and \( Q(-5,-5) \):
- \( x_1 = 2 \), \( y_1 = 9 \)
- \( x_2 = -5 \), \( y_2 = -5 \)

Calculating the gradient \( m \):

\[
m = \frac{-5 - 9}{-5 - 2} = \frac{-14}{-7} = 2
\]

### 3. Points D (1,3) and E (4,-3)

Let \( D(1,3) \) and \( E(4,-3) \):
- \( x_1 = 1 \), \( y_1 = 3 \)
- \( x_2 = 4 \), \( y_2 = -3 \)

Calculating the gradient \( m \):

\[
m = \frac{-3 - 3}{4 - 1} = \frac{-6}{3} = -2
\]

### Summary of Gradients

1. Gradient of line AB = \( \frac{5}{2} \)
2. Gradient of line PQ = \( 2 \)
3. Gradient of line DE = \( -2 \)

### Graphical Representation

To represent these lines graphically:

1. **Line AB:**
- Points A(2, 5) and B(-2, -5)
- The slope is \( \frac{5}{2} \), indicating a steep positive slope.

2. **Line PQ:**
- Points P(2, 9) and Q(-5, -5)
- The slope is \( 2 \), indicating a positive slope but less steep than line AB.

3. **Line DE:**
- Points D(1, 3) and E(4, -3)
- The slope is \( -2 \), indicating a negative slope.

### Graph

You can graph these points and lines using graphing software or graph paper:

1. **For line AB**, plot (2, 5) and (-2, -5) and draw a line through them.
2. **For line PQ**, plot (2, 9) and (-5, -5) and draw a line through them.
3. **For line DE**, plot (1, 3) and (4, -3) and draw a line through them.

Each line will show the steepness of the gradient, with slopes represented visually.