Harder Sketching

This page is not really year Eight stuff, but I’ve written it knowing that (i) some years we go a bit further (ii) a few people will want to know more (iii) a few older pupils (and parents) will want to know what the higher skills are. This page should get you going, and I might well add a page to the Sixth Form stuff along the same lines.

Basic questions to consider when looking at sketching a function:

Do I recognise this type / class of function?

Where is x zero?

Where is y zero?

Are there any (other) interesting places?

Where is the gradient zero?                                                       [called a turning point]

Does something interesting happen as x or y go to infinity?       (called an asymptote)

Does the curvature change from ) positve  to ( negative ?             [called an inflexion]

Let’s look at those slowly:

An explicit function means y is some expression in x, such as y = x/(2-x), y = sin(x+2). An implicit function has x and y mixed up, such as xy = 6, sin(xy) - x = 1. Generally we work with explicit functions or ones we can easily rewrite as explicit.

The type or class of function: you might well recognise some of these

        Straight lines    ax + by = c, y = mx + c

        Parabolas        y = x² , y = ax² + bx + c            Parabola means ‘like a ball’

        Hyperbolas        y = a/x,     xy = c,

        Circles                x² + y² = r²                             and   Ellipses              ax² + by² = c²

        Cubics                y =  x³      y = ax³ + bx² + cx + d

Where is x or y zero?   With harder expressions this requires some algebra, as in y = (x-2)(x+3); sometimes it answers one of the later questions, as in y = 2 + 1/x   or   xy = 6

Are there any (other) interesting places? Some of the answers to this have probably been found by now, but,  for example, x = 2 is interesting in any expression which includes (x-2) in it - because it gives a zero value. 

Where is the gradient zero?  Heading towards seriously hard maths; you can find a (usually slightly less difficult) formula for the gradient of a curve: the simple version says that every axn term in your base function becomes an anxn-1 . When this gradient function is zero, [the first differential]. you have a turning point.

Does something interesting happen as x or y go to infinity? Easier to spot and valid maths for Middle school, you’re looking for values of x that make y infinite, such as x=2 in y = 1/(2-x) or for trends such as y never getting over 6 in y = 6 - 1/(2-x) while the size of x is bigger than two.

Does the curvature change sign? Hard maths: the simplest version of this says that the tangent crosses the line; the slightly harder way (that finds the place of change) says that every ax term in your base function becomes of the form an(n-1)xn-2 . When this function is zero, [the second differential]. you have an inflexion. The inflexion just might also be a turning point; lots of the Sixth form are confused by this idea.

2013 additions, having returned to teaching middle and upper school.

Higher level GCSE students will recognise y=1/x and be able to sketch it in the air (Air Maths, like Air Guitar-playing); it is a curve that, in the positive quadrant, comes down the y-axis and creeps off along the x-axis, also symmetrical around x+y=0. It has asymptotes on both axes.  The same students will not so easily recognise y-3 = 1/(x-2), which is the same curve, but with the asymptotes moved to centre on (2,3). The (usually good Y12) students who do ‘see’ this often show incomprehension (perhaps feigned) at the lack of vision of their colleagues. I suggest that there is a moment within the reading where some equivalent to ‘standing back from a painting to try to see the whole’ applies to the reading of the equation.

Also, a recent example from an adjacent paper in the same syllabus was something like y=4x+8 + 1/(x-2).   While x is large, this curve is trying to be y=4x+8, a nice (easy) straight line; the reciprocal 1/(x-2) is irrelevant. When x gets close to 2, though, the 4x+8 becomes irrelevant and the curve is trying to be the y=1/x curve, centred on the asymptote at x=2. A little checking to see what happens each side of x=2 (x<2 is negative, x>2 is positive) and the curve is ready to sketch. Yes; it is that easy – once you ‘see’ (or are persuaded) how the elements go together.

© David Scoins 2017