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  • 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:

Fidning gradient of a striagh line. Linear graph gradient. Gradient. Straight line graph. GCSE Maths, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

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):

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  • 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.

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