The problem asks to determine whether the gradient of each given line is positive, negative, zero, or undefined, and then to calculate the gradient.

1. Problem Description

The problem asks to determine whether the gradient of each given line is positive, negative, zero, or undefined, and then to calculate the gradient.

2. Solution Steps

Graph 1:

(a) The line slopes upwards from left to right, so the gradient is positive.

(b) We can pick two points on the line to calculate the gradient. Let's choose (0, 0) and (1, 1).

The formula for the gradient $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points (0, 0) and (1, 1) into the formula:

$m = \frac{1 - 0}{1 - 0} = \frac{1}{1} = 1$

Graph 2:

(a) The line slopes downwards from left to right, so the gradient is negative.

(b) We can pick two points on the line to calculate the gradient. Let's choose (0, 1) and (2, -1).

The formula for the gradient $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points (0, 1) and (2, -1) into the formula:

$m = \frac{-1 - 1}{2 - 0} = \frac{-2}{2} = -1$

Graph 3:

(a) The line appears to be horizontal, thus the gradient is zero.

(b) We can pick two points on the line to calculate the gradient. Let's choose (0, 2) and (1, 2).

The formula for the gradient $m$ is:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the points (0, 2) and (1, 2) into the formula:

$m = \frac{2 - 2}{1 - 0} = \frac{0}{1} = 0$

3. Final Answer

Graph 1:

(a) Positive

(b) 1

Graph 2:

(a) Negative

(b) -1

Graph 3:

(a) Zero

(b) 0