Sec 3 E-Math Functions & Graphs notes

The Cartesian plane locates every point by an ordered pair (x, y). A linear function y = ax + b pairs every x with exactly one y, and its graph is a straight line. The number a is the gradient — how fast y climbs for each step of x — and b is the y-intercept, where the line crosses the y-axis. Two points are enough to draw the whole line.

1. 1Plot two or three points

   For y = 2x + 4: (0, 4), (1, 6), (2, 8). A third point is a free accuracy check.

2. 2Join with a ruler

   Extend across the grid; label the line with its equation.

3. 3Read features off the graph

   Crosses the y-axis at (0, 4); crosses the x-axis where y = 0: x = −2.

Special cases: a horizontal line such as y = 5 has gradient 0 (no rise at all), and a vertical line such as x = −3 has an undefined gradient (the run is zero). Parallel lines have equal gradients, so anything parallel to y = 2x + 4 also has gradient 2.

Between points (x₁, y₁) and (x₂, y₂), the gradient is (y₂ − y₁)/(x₂ − x₁) — vertical change over horizontal change. Subtract the coordinates in the same order on top and bottom, or the sign will betray you. A positive gradient climbs left-to-right; a negative gradient falls; a bigger number (ignoring sign) means a steeper line.

• Through (1, 3) and (5, 11): (11 − 3)/(5 − 1) = 8/4 = 2 — climbing.
• Through (1, 7) and (3, 1): (1 − 7)/(3 − 1) = −6/2 = −3 — falling steeply.
• y = 4x − 7 already shows its gradient (4) and y-intercept (−7) by inspection.

Equations sometimes hide the gradient: for 2y = 6x − 4, divide every term by 2 first to get y = 3x − 2, and only then read off gradient 3 and intercept −2. Dividing one term only is a classic error.

A quadratic function y = ax² + bx + c (a ≠ 0) draws a smooth U-shaped curve called a parabola. The sign of a decides the shape: a > 0 opens upward and the turning point is a minimum; a < 0 opens downward with a maximum. The curve is symmetric about a vertical line of symmetry through its turning point.

• Table of values: substitute x = −2, −1, 0, 1, 2… into y = x² − 2x − 3 to get 5, 0, −3, −4, −3… — the repeating values expose the symmetry.
• Factorised form y = (x − a)(x − b): the graph cuts the x-axis at x = a and x = b; the line of symmetry runs halfway between them.
• Completed square y = (x − p)² + q: the turning point sits at (p, q) — a minimum if the bracket is positive-squared, read together with the sign of a.

Where a curve and a line meet, their equations are satisfied together: set them equal and solve. The x-values found are the x-coordinates of the intersection points, and substituting back gives the y-values.

A power function y = axⁿ multiplies a power of x. The syllabus names six exponents, and each exponent has a signature shape: n = 1 draws the straight line y = ax; n = 2 the familiar parabola; n = 3 an S-shaped cubic through the origin; n = 0 the horizontal line y = a; n = −1 the hyperbola y = a/x with two branches that approach but never touch the axes; n = −2 a similar two-branch shape hugging the axes more tightly.

An exponential function y = kaˣ (with a a positive integer, a ≠ 1) multiplies repeatedly: each step of x multiplies y by a. Every graph of the family passes through (0, k) — in particular aˣ always passes (0, 1) because a⁰ = 1 — climbs ever more steeply for a > 1, and never dips below the x-axis.

Sums of at most three power functions, such as y = x² + 2x or y = x³ − x, combine the shapes: the highest power dominates far from the origin, while lower powers bend the curve near it. Tables of values remain the reliable way to plot these hybrid curves.

Unlike a straight line, a curve changes its steepness from point to point. The gradient of a curve at a point is estimated by drawing the tangent — the straight line that touches the curve at that point and follows its direction — then calculating the tangent’s gradient from two convenient points on it.

1. 1Place your ruler at the point

   Adjust its angle until it just touches the curve there without cutting across it.

2. 2Draw the tangent

   Extend it across the grid so you can pick two far-apart points on it.

3. 3Read two lattice points on the tangent

   For example (0, 1) and (2, 9).

4. 4Compute the gradient

   (9 − 1)/(2 − 0) = 4 — the estimated gradient of the curve at that point.

Because tangent drawing is by eye, mark schemes accept a range of gradient values — but the method must be sound: one clear tangent, two clearly marked points on it, and gradient = rise ÷ run. Points used must lie on the tangent, not on the curve.