Year 9 Top Set: Linear Graphs Mastery

Gradient and Equation of a Straight Line

• The gradient of a straight line through points (2, 3) and (4, 7) is calculated as the change in y over the change in x, resulting in a gradient of 2.
• The equation of a straight line through point (3, 2) with a gradient of 2 is expressed as y = 2x - 4 after applying the point-slope formula.
• For the points (1, 3) and (2, 5), the gradient is 2, leading to the equation of the line being y = 2x + 1.

Properties of Gradient

• A straight line parallel to y = 3x - 2 will have the same gradient of 3; therefore, the gradient of a line through (2, 5) that is parallel to this line is also 3.
• The gradient of a line perpendicular to the line represented by y = 2x + 1 is the negative reciprocal of 2, resulting in a gradient of -1.

Characteristics of Lines and Gradients

• The equation y = 2x - 3 demonstrates a positive gradient of 2, emphasizing that a negative gradient would indicate a downward slope.
• Perpendicular lines feature gradients that multiply to -1, confirming the relationship between their inclinations.
• A line with a gradient of 0 is horizontal, indicating no vertical change regardless of horizontal movement.
• The equation x = 2 represents a vertical line with an undefined gradient, characteristic of vertical lines where x remains constant while y varies.