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Notes by Category University Engineering

Mechanics & Stress Analysis*
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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, 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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If you do not know the coefficient, then you can find out which way the graph goes by seeing what happens as x tends to ±∞:

  • If as x ∞, y and x → -∞, y → -, the graph is positive

  • If as x ∞, y → - and x → -∞, y , the graph is negative

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Cubic graphs can have just 1 or 3 distinct roots, 1 distinct root with a repeated root, or 1 triple repeated root.

  • A triple repeated root occurs when the graph has just one stationary point, and this is on the x-axis

  • A distinct root occurs when the graph crosses the x-axis

  • A repeated root occurs when the graph touches the x-axis but does not cross it

To sketch a cubic, you need to know the roots. If it is given in the form ax³ + bx² + cx + d, you need to factorise it first. This will tell you how many roots it has, and where they are. Then, testing to see what happens as x tends to ±∞ shows the shape.

Quartic Graphs

The standard form for a quartic function is ax⁴ + bx³ + cx² + dx + e where a, b, c, d and e are real numbers and a is not zero.

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Again, you need to know the roots of the function and the y-intercept to be able to sketch it.

Reciprocal Graphs

To sketch graphs of reciprocals, such as y = 1/x, y = 1/x², or -3/x, you need to know the asymptotes. These are lines that the graphs tend towards, but never touch or cross.

Graphs in the form y = k/x or y = k/x² have asymptotes at x=0 and y=0

The greater the value of the numerator, the further the graph is from the coordinate axis. The asymptotes are still y=0 and x=0, however.


In maths, functions are relationships that map a value from a set of inputs to a single output. The set of inputs is known as the domain, and the set of possible outputs is the range. The roots of a function are the values of x for which f(x) = 0.

There are two types of functions, one-to-one and many-to-one. Anything else is not a function:

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Composite Functions