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The University of New South Wales School of Mathematics and Statistics Mathematics Drop–in Centre
GRAPHS
Graph sketching is a very important skill. From a well drawn
is increasing or decreasing.tion, including roots, limits, turning points and where the functiongraph you may be able to immediately see properties of a func-
Graphs should always be
large
and
neatly drawn
, and important features should be
labelled
x
y
x
y = 6
23
The equation
ax
(^) by
c represents
a
straight line
Often the easiest way
sider 3tercepts on the axes. For example, con-to sketch its graph will be to find its in-
x
y
= 6.
Substituting
x
= 0
and solving for
y
gives the
y –intercept
23 (^) ); letting
y
= 0 gives the
x –intercept
(^) 0); we plot these points and draw the line joining them. 0
x
y y = x 2 − 5 x
25
The quadratic equation
y
ax
2 +^
(^) bx
(^) c
represents a
parabola
To sketch the
accurate sketch theneed) and note its concavity. For a morerevision worksheets on quadratics if yougraph we need only find its roots (see
y –intercept and ver-
ample,tex may also be useful. Consider, for ex-
y = x 2 − 5 x
There are
x
x intercepts at the roots of the quadratic,
= 1 and
x
= 4, and since
x 2 has a pos-
itive coefficient the parabola is concave upwards. The
y –intercept
is
y = 4. The vertex has
x –coordinate halfway between the roots,
that is, at
x
=
25 (^) , and by substituting this into the quadratic we
obtain the
y –coordinate
y
=
−
4 9 (^).
x
y
x (^) + 2)
2
y −
2 = 9
The graph of
x 2 + y 2 = r
2
is the
circle
having centre at the origin and
radius
r ; if the centre of the circle is at
x = a , y = b
instead, then its equation
is (
x
−
a ) 2
y −
b ) 2
=
r 2 .
For an
example of the latter,
x
2
y −
2 = 9
is the equation of a circle with centre (
(^) 1) and radius 3.
x
y
x 2
y 2 = 18
The equation of an
ellipse
is similar
(positive) coefficients on theto that of a circle, but with different
x 2
and
y 2
terms.
The standard form of the
equation is
x 2
a 2
y 2
b 2
= 1
for which the ellipse is centred at the
origin, and has semi–axis lengths
a
in the
x
direction and
b
in
the
y
direction.
The best way to deal with an equation such as
x 2
y 2
= 18 is to divide both sides by 18 in order to put the
equation into standard form, (
x 2 /
y 2 /
- = 1.
This is an
ellipse having semi–axis lengths
18 and 3.
x
y 5 x 2 − y 2
y
=
√
5 x
If in the previous case the
y 2
term has a
negative coefficient, the curve is a
hyperbola
x 2
a 2 −
y 2
b 2
= 1
by ( To sketch it, first draw the asymptotes given
x 2 /a
2 ) −
( y 2 /b
2 ) = 0, that is,
y
=
bx/a
example, 5then place the two branches of the curve. For
x 2 −^
(^) y 2 = 10 has as asymptotes the
lines
y = ± √ 5 x
; it has
x –intercepts at
and no
y –intercepts.
EXERCISES
Please try to complete the following exercises.
Remember that
you
cannot
expect to understand mathematics without doing lots
of practice!
Please do not look at the answers before trying the
please consult your tutor or the Mathematics Drop–in Centre.which you cannot find, or a question which you cannot even start,working carefully, find the mistake and fix it. If there is a mistakequestions. If you get a question wrong you should go through your
- Draw
large
neat
labelled
graphs of the following.
(a) 5
x (^) + 4
y
= 32;
(b)
y
= 2
x 2 − 4 x −
(c)
x 2
y 2 = 49;
(d)
y = − x 2 −
x
(e)
x 2
y 2
(f)
x 2
y 2
(g) 2
y
=
x
(h) 3
x 2
y 2 = 48;
(i) (
x −
2
y
2 = 16;
(j)
x 2 −
y 2 = 25.
- The equations (
) can be modified to give ellipses and hy-
way as we did for circles.perbolas with centres away from the origin in much the same
Find the centres of the following
curves, and sketch them. (a)
x
− (^) 1)
2
y
2
= 1; (b) 5
x 2 −
( y −
(^) 2)
2 = 10.
- What is the difference between the hyperbolas we have seen
and one in which the coefficient of
x 2 , instead of
y 2 , is nega-
tive? Sketch (a)
− 5 x 2 + y 2
(b) 4
y 2 −
(^) x 2 = 16.
- The equation
xy
= constant also defines a hyperbola, but it
is positioned differently from those above. Sketch (a)
xy
(b)
xy
What are the asymptotes of these curves?
ANSWERS
. In these answers we have
described
the graphs set
in the exercises, but of course you need to actually
draw
them!
(a) A line with intercepts (
(^32) 5 , (^) 0) and (
(b) A parabola, concave upwards,
x –intercepts at
1 and 3,
vertex at (
- and
y –intercept at
(d) A parabola, concave downwards,(c) A circle, centre at the origin, radius 7.
x –intercepts at
7 and
5, vertex at (
(^) 36) and
y –intercept at 35.
(e) An ellipse, centre at the origin, semi–axis lengths 2 in
the
x
direction and 5 in the
y
direction.
(f) A hyperbola, centre at the origin, asymptotes
y
=
3 4 (^) x
and
x –intercepts
(g) A line with
x –intercept
6 and
y –intercept 3.
(h) An ellipse, centre at the origin, semi–axis lengths 4 in
the
x
direction and
6 in the
y
direction.
(i) A circle with centre (
- and radius 4.
Note
for your
sketch: the circle will have the
y
axis as a tangent.
(j) A hyperbola with centre at the origin; the asymptotes
are
y
x
and the
x –intercepts
(a) An ellipse, as on page 2 but shifted to have centre (
(b) A hyperbola, shape exactly as on page 2 but shifted to
have centre (
(^) 2). The asymptotes are
y −
(^) 2 =
(^) x .
- The branches of the hyperbolas will be located at the “top
and bottom” of the plane instead of “left and right”.
- The asymptotes will be the
x
and
y
axes.
Curve (a) is a
and fourth.hyperbola in the first and third quadrants, (b) in the second
Turn your paper through 45
◦
to see that they
are the same shape as in the text and previous exercises.
