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.
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
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.
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.
Again, you need to know the roots of the function and the y-intercept to be able to sketch it.
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:
Two functions can be combined to form a composite function:
fg(x) = f(g(x)) Apply g first, then apply f to this
Piece-wise Defined Functions
Often, functions will be split up into two or more parts, each of which applies for a certain range of values.
The modulus of a number is its non-negative (or absolute) numerical value. For example |-3| = 3.
The modulus of a function, therefore, is function where all input values give a positive output, regardless of whether or not the input (or x-value) is positive or negative:
For a modulus functions y = |f(x)|:
When f(x) ≥ 0, |f(x)| = f(x)
When f(x) < 0, |f(x)| = -f(x)
This is easiest shown on a graph of y=x:
However, it is also possible to have the function of a modulus, rather than the modulus of a function. This is noted as y = f(|x|), not y = |f(x)|, and represents a reflection in the y-axis:
It is important not to get confused between y = |f(x)| and y = f(|x|)
The inverse of a function, f‾¹(x) does the exact opposite to the original function, f(x) - it maps the range of the original function to its domain. Since functions cannot be one-to-many, inverse functions can only be one-to-one.
f(x) and f‾¹(x) are inverses of each other
ff‾¹(x) = x
The domain of f(x) is the range of f‾¹(x)
The range of f(x) is the domain of f‾¹(x)
The graphs of f(x) and f‾¹(x) are reflections of each other in the line y=x
There are a number of different types of graph transformations, that move every single point on a graph by a certain amount in a certain way.
Transformations can be expressed as functions, or as vectors.
When multiple transformations are combined, do one after the other. It is generally helpful to sketch out each individual transformation on a separate axis to avoid getting confused.