Mr Kidman's Classroom
Revise the key ideas
Coordinates and gradients
- Coordinates (x, y) give horizontal position first, then vertical position. In quadrant II (the upper-left quarter of the coordinate plane), x is negative and y is positive; the origin is (0, 0).
- The midpoint of (x₁, y₁) and (x₂, y₂) is (, ). The midpoint of (−2, 3) and (6, 7) is (2, 5).
Add the two x-coordinates and divide by 2. Do the same with the two y-coordinates. Each result lies halfway between the corresponding coordinates.
- A straight line has constant gradient m = . Use two well-separated points on the line; a positive gradient rises to the right.
Graph of y equals two x plus one - Worked example: Through (1, 3) and (4, 9), m = = 2. Subtract the coordinates in the same order in numerator and denominator.
- In y = mx + c, m is the gradient and c is the y-intercept, where the line meets the vertical y-axis. y = 2x + 1 passes through (0, 1), and goes up 2 for every 1 across.
- To plot a line, calculate a table of values and draw a straight line through the points. Check the intercept and a third point to catch errors.
- A horizontal line is y = k and has gradient 0. A vertical line is x = k; its gradient is undefined because the horizontal change is zero.
Finding line equations
- Worked example: A line with gradient 3 through (2, 5) has 5 = 3 × 2 + c, so c = −1 and its equation is y = 3x − 1.
- For a line through two points, find the gradient first, then substitute either point to find c. Through (1, 3) and (4, 9), y = 2x + 1.
- Parallel non-vertical lines have equal gradients. y = 2x + 1 and y = 2x − 5 are parallel; different intercepts make them distinct lines.
- Rearrange another line form to identify gradient: 2y = 6x − 4 becomes y = 3x − 2, so the gradient is 3, not 6.
Higher — perpendicular lines
- For perpendicular non-horizontal, non-vertical lines, gradients satisfy m₁m₂ = −1. Take the negative reciprocal: the line perpendicular to gradient 2 has gradient −.
- Worked example: Perpendicular to y = 2x + 1 through (4, 3): y = −x + c. Substitution gives 3 = −2 + c, so y = −x + 5.
- A horizontal line is perpendicular to a vertical line. The negative-reciprocal rule should not be used by trying to divide by a zero gradient.
