Graphs, Sequences and Variation: notes

Straight Lines

• Gradient: rise / run = (y2 - y1) / (x2 - x1). Between (1, 2) and (3, 8) the gradient is 6 / 2 = 3.
• Equation: y = mx + c, where m is the gradient and c is where the line crosses the y axis.
• Parallel lines: same gradient. Perpendicular lines: the gradients multiply to -1.
• Midpoint: the average of the coordinates. Midpoint of (2, 4) and (6, 10) is (4, 7).

Curves and Travel Graphs

• Quadratic graph: a U shape (or an upside down U) with a turning point and a line of symmetry.
• Roots: where a graph crosses the x axis, y = 0.
• Travel graph: distance against time. A flat line means stopped. The gradient is the speed.
• Speed-time graph: the gradient is acceleration. The area under the graph is distance.

Sequences

• Linear sequence: add the same number each time. 3, 7, 11, 15 has nth term 4n - 1.
• nth term: the common difference is the number in front of n; then adjust for the first term.

Variation

• Direct variation: y = kx. Doubling x doubles y.
• Inverse variation: y = k / x. Doubling x halves y.
• Method: use the given pair of values to find k first.