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