10.2.4 Curve sketching in Cartesian coordinates
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291 310➤Curve sketching is exactly that – it does not meanplottingthe curve, as in Chapter 3,
although one might actually plot a few special points of the curve, such as where it
intercepts the axes. A sketch of a curve shows its general shape and main features, it is
not necessarily an accurate drawing. In sketching a curve we deduce what we can about it
from quite general observations, such as where it increases, decreases, or remains bounded
asxbecomes large or small. Similarly we might look for stationary values, intercepts on
the axes and so on.
With modern calculators capable of plotting almost any curve we might wonder why
curve sketching is necessary at all – is it like shoeing horses, for example? Not at all.
Apart from the fact that you might not always have a graphics calculator to hand, the
main benefit of the skills of curve sketching is that it gives you an appreciation of how
functions behave, and what the properties of the derivative can tell you about this. It
gives you a ‘feel’ for the function. Being able to sketch the rough shape of a given
rational function, for example, is a far more portable skill than being able to graph it on
a calculator by pressing a few buttons.
Of course, we already know a large number of different graphs of the various ‘elementary
functions’ we have considered, and the first step in sketching a curve is to see whether it
is easily converted to one with which we are already familiar. The kinds of transformation
we can make are listed below.
1.y=f(x)+ctranslates the graph ofy=f(x)bycunits in the direction 0y.
2.y=f(x+c)translates the graph ofy=f(x)by−cunits in the 0xdirection.
3.y=−f(x)reflects the graph ofy=f(x)in thex-axis.
4.y=f(−x)reflects the graph ofy=f(x)in they-axis.
5.y=af (x )stretchesy=f(x)parallel to they-axis by a factora.
6.y=f(ax)stretchesy=f(x)parallel to thex-axis by a factor 1/a.
However, once these transformations are taken advantage of, we might still have quite
complicated graphs to sketch. We can follow a systematic procedure for sketching graphs
which can be nicely summarised in the acronymS(ketch) GRAPH:
- Symmetry – does the curve have symmetry about thex-ory-axes, is
the function odd or even? - Gateways – where does the curve cross the axes?
- Restrictions – are there any limits on the variable or function values?
- Asymptotes – any lines that the curve approaches as it goes to infinity?
- Points – any points of special interest which are worth plotting?
- Humps and hollows – any stationary points or points of inflection?
(I am indebted to Peter Jack for most of this pretty acronym.)
Working through each of these (not necessarily in this order) should provide a good
idea of the shape and major features of the curve. We describe each in turn.
S.We look for anysymmetryof the curve. For example, if it is an even function,
f(x)=f(−x)then it is symmetric about they-axis,andweonlyneedtosketch