Page 23 - Math SL HB Sem 2

Curve sketching

You will often need to draw a rough sketch of a curve without
plotting a large number of points. Here are five steps to follow
when sketching a curve.
.1. Zcros. Find th-e value of y when :r : 0 and find (if possible)
the value(s) ofr when :
7 0. This gir.es the points where the
curve crosses the axes.
.i.
Infinities Find the values of x for which y is not defined.
In most cases these u,ill be values of x which make the
denominator of a rational function zero.
.1.
Sign There are two places where a curve might change sign:
:
i) at y 0 (where the curve crosses the x-axis)
n) at y: co (at infinities).
d.y
* Turning points Calculate
dx'
d.t
When Z ) 0, the curve slopes upwards from left to right.
dJc
dv
When (
* 0, the curve slopes downwards from left to right.
dv
When = 0, the curve has a turning point.
Or
ii Aqrmptotes Examine the behaviour of the function aS
: x-+ +oo and aS x --1 -oo.

E(ample 4
ftn"d ?3 2$,
:
Sketch tlre curve , '* 1
' 1- )'
-
t,
u) f.=O t b'- 13
V-^ .,--l -.-'*----.'-'.--r)
.r_ v r r -
-t
!) )x - ?= o
v.}/t [v,,]) 3
.) ,6a" A I /:c', x. -l i

Y. oo , t -- >/z' : !,

J) d,) t
(:x -3)

A9 <0 dtt^4! Y-4t*.- Loa-6

\+€
e) A$,a^^}f . ! 'l
?l+ I ['^^ t+1.
-,u ,'vr 2* -j x-"10q 7-?,t L
* -') 4)