- A-Level Maths

# Graphs, Functions & Transformations

When sketching graphs, it is important to clearly show and label any coordinate-axis intercepts (y-intercepts and roots) as well as any stationary points (e.g. turning points).

### Linear Graphs

The general from for a linear graph is ** y = mx + c**, where m is the gradient and c the y-intercept. Gradient is found as rise/run:

This equation can be rearranged to give an alternate equation for a line, which is more useful when you know two points and need to know the line connecting them.

y2 - y1 = m(x2 - x1)

To find the length of a section of line, use **Pythagoras' Theorem.**

Two

**parallel lines**have an equal gradient, so will never meet.Two

**perpendicular lines**have gradients that are each other's negative reciprocal, and so they do cross. This means that**the product of their two gradients equals -1**

### Quadratic Graphs

The general form of a quadratic expression is **ax² + bx + c.**** **All quadratic graphs are parabola-shaped, symmetrical about one turning point (this can be a maximum or minimum):

For quadratics in the form

*ax² + bx + c*,**c****is the y-intercept.**Completing the square gives the coordinates of the turning point:

**When***f(x) = a(x + p)² + q*, the turning point is at (-p, q)The discriminant tells you how many roots there are, so how many times the graph crosses the x-axis.

### Cubic Graphs

The general form for a cubic expression is * ax³ + bx² + cx + d*, and can intercept the x-axis at 1,2 or 3 points.