x Graph Sketching (Sixth, rapidly)

Start from recognising that you are NOT plotting a graph. You are doing something different, called sketching. I shall assume that you have a function y = f(x) to plot. 

You are assumed to have already read—or to not need at all—the similar pages in Lower School under Geometry.

For sixth formers and those pushing the calculus boundary early: The first differential is dy/dx and is understood to be zero at stationary values; these can be maxima, minima or points of inflexion. The second differential is zero at points of inflexion – these may also (simultaneously) be stationary values. A sketch identifies the asymptotes, the intercepts, the turning points, intersections with other curves – and the basic shape of the curve.


Routine questions:
Do you recognise the type of curve the function represents?
What happens when x or y is zero? This will give you the (axis) intercepts.
Are there any obvious values of x or y to look at? (e.g. a term of (x-3) strongly suggests you should look what happens when x=3).
What happens as x or y becomes very large (tend to ± infinity)? This should give you any asymptotes (especially the ones you haven’t already found).
Where are the turning points? (i.e. differentiate the function, equate this differential to zero and solve – or try to) What about points of inflexion (i.e. where the second differential is zero)?


Can you apply a simple transformation that lets you sketch the curve more easily? Look first for translations. Then try stretches. Rotations are hard to work out until you are good with matrices.
Is the curve odd or even (or neither)? 
For an even function,  f(-x) = f(x)  so the curve is symmetrical about the y-axis; 
for an odd one f(-x) = - f(x) so there is a rotation about the origin of 180º.


Functions you should work on recognising:
The straight line         y = mx + c  or ax + by = c
The parabola                y = ax
² + bx +c
The hyperbola                y = a/x

These three are (technically) covered at GCSE, possibly even at Intermediate level. At Higher level and beyond you can add these:
The circle                    (x-a) ² +(y-b) ² = r²   has centre (a,b) and radius r
The exponential         y = a e
kx goes thro’ (0,a), all y>0 asymptote x=0 for y<<0
The logarithm             y = a
-1 ln (x/k)the inverse function of the exponential abov
The sine curve            y = sin x


A translation of (a,b) replaces x with (x-a) and y with (y-b), so x 2 + y 2 = r2 has centre (0,0) but (x-a) 2 +(y-b) 2 = r2 has centre (a,b).

A one way stretch in the x-axis of scale factor p will replace x with x/p,

so  y = a sin kx could be seen as y/a = sin kx, which is a two-way stretch of y = sinx, by scale factor a in the y-axis and by a factor 1/k in the x-axis. Check this new function goes thro’ (0,0), (π/2k, a),  (3π/2k, -a), (2π/k, 0). The cosine function is a translation by π/2 (in the x-axis, positively) of the sine function, since cos(x-π/2) = sin x.


Examples to consider plotting:

Y = 3(x+2) + 5/(x+2)2                              ….                      25(x-a) 2 +(4y-8b) 2 = 64

Y = 2 cos (3x-π/6) + 1                           ….                       y – 6 = 1/(x-4)


Not Covered Here:

Curves defined in more than two dimensions
Curves defined in polar co-ordinates
Curves in complex space



Related pages: Year Eight basics; Year Eight Hard Stuff—if any of what is above is difficult then you really should go back a stage or two. Don’t be put off by the title, you’ll go a lot faster than back then.


General:

The ability to sketch graphs may be reduced as calculators become more sophisticated. It is certainly true that we use pictures more than we did. I suspect that sketching will become one of the non-calculator skills as it demonstrates very quickly the level of understanding achieved by a pupil. It that sense it is worth working on. I have written notes (‘Sketching’ is in the title line) that cover sketching to Advanced level. 

Really, though you just might want more at University.

Your computer may have a grapher package. If so, the examples may open a world of fascination. Beware, though, playing with these causes time to vanish inexplicably...

© David Scoins 2017