## Harder Sketching 2

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      y2 = 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 says that the tangent crosses the line; the slightly harder way (that finds the place of change) says that every axn 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.