CURVE SKETCHING
S.No
Polynomial form
Factorized form
Completed square form
Easy to find y-crossing
Parabola
Easy to find
Easy to find the turning
point
Set
point.
If c>0, U-shape Parabola
If c<0, ∩-shape Parabola
to find turning Easy to find x-crossings
Turning point co-ordinates
(-a,b)
If p>0, U-shape Parabola
If p<0, ∩-shape Parabola
Example:
---
Easy to find y-crossing
Easy to find
Set
points
Cubic
Graph
to find turning
--Easy to find x-crossings
If p>0, ∩U -shape cubic
If p<0, U∩ -shape cubic
Reginald Lourd Raj R
Page 1 of 7
Curve Sketching / 0580 / Mathematics
A cubic equation can intersect the x-axis up to three times.
y-intercept (y-axis crossings): Substitute x = 0 in the equation and solve for y
x-intercept (x-axis crossings): Substitute y = 0 in the equation and solve for x
Addition of constant-term: The effect of adding a constant term is to shift the graph upwards by the same
constant. (The effect of subtracting a constant term is to shift the graph downwards by the same constant).
Reginald Lourd Raj R
Page 2 of 7
Curve Sketching / 0580 / Mathematics
ELEMENTARY PARABOLAS AND CUBIC GRAPHS
Lines of Symmetry:
Reginald Lourd Raj R
Lines of Symmetry:
Page 3 of 7
Curve Sketching / 0580 / Mathematics
RECIPROCAL FUNCTIONS
The graph consists of two separate branches, you must not join them together.
When the equation is in the form
without ever touching them.
, the curve approaches both axes and gets closer and closer to them
An asymptote is a line that a graph approaches but never intersects.
Asymptotes: x-axis and y-axis
Lines of Symmetry:
and
Asymptotes: x-axis and y-axis
Lines of Symmetry:
and
Asymptotes: x-axis and y-axis
Lines of Symmetry:
Asymptotes: x-axis and y-axis
Lines of Symmetry:
Reginald Lourd Raj R
Page 4 of 7
Curve Sketching / 0580 / Mathematics
EXPONENTIAL FUNCTIONS
(In all cases below, b > 1)
Asymptotes: x-axis
Lines of Symmetry:
Asymptotes: x-axis
Lines of Symmetry:
Asymptotes: x-axis
Lines of Symmetry:
Asymptotes: x-axis
Lines of Symmetry:
Reginald Lourd Raj R
Page 5 of 7
Curve Sketching / 0580 / Mathematics
TRIGONOMETRIC FUNCTIONS
The graph repeats itself every 360° in both the positive and negative directions.
Notice that the section of the graph between 0 and 180° has reflection symmetry, with the line of reflection
being x = 90°. This means that sin x = sin (180° - x ), exactly as you should have seen in the investigation
above.
It is also very important to notice that the value of sin x is never larger than 1 nor smaller than -1.
The graph repeats itself every 360° in both the positive and negative directions.
Here the graph is symmetrical from 0 to 360°, with the reflection line at x = 180°. This means that
cos x = cos (360° - x ), again you should have seen this in the investigation above.
By experimenting with some angles you will also see that cos x = - cos (180° - x )
It is also very important to notice that the value of cos x is never larger than 1 nor smaller than -1.
Reginald Lourd Raj R
Page 6 of 7
Curve Sketching / 0580 / Mathematics
The vertical dotted lines ( x = 90˚ and x = 180 ˚ ) are approached by the graph, but it never touches nor
crosses them. They are the asymptotes.
Notice that this graph has no reflection symmetry, but it does repeat every 180°. This means that
tan x = tan(180° + x ).
Note that, unlike sin x and cos x, tan x is not restricted to being less than 1 or greater than -1.
Reginald Lourd Raj R
Page 7 of 7
Curve Sketching / 0580 / Mathematics
6
E2.11 Recognise, sketch and interpret graphs of functions
3
y
O
y
x
O
A
B
y
y
O
x
O
C
D
y
y
O
x
E
O
x
x
x
F
Write down the letter of the graph which could represent each of the following equations.
(a) y = 2 – x2
(b)
............................................ [1]
y = 2–x ............................................ [1]
2
(c) y = x © UCLES 2020
............................................ [1]
0580/PQ/20
7
4
Sketch the graph of 2x + 3y = 18 .
On your sketch, write the values where the graph crosses the x-axis and the y-axis.
y
O
© UCLES 2020
x
[2]
0580/PQ/20
8
5
Sketch the graph of y = x2 + 2x .
On your sketch, write the values where the graph crosses the x-axis and the y-axis.
y
O
x
[3]
© UCLES 2020
0580/PQ/20
9
6
y
30
–5
–2 0
NOT TO
SCALE
x
The diagram shows a sketch of the graph of y = ax2 + bx + c .
Find the values of a, b and c.
a = ............................................... [5]
b = ............................................... [5]
c = ............................................... [5]
© UCLES 2020
0580/PQ/20
10
7
Sketch the graph of y = x2 – 3x – 10 .
On your sketch, write the coordinates of any turning points and the values where the graph crosses the
x-axis and the y-axis.
y
O
x
[7]
© UCLES 2020
0580/PQ/20
11
8
(a) Solve the equation (x – 5)(2x2 – 18) = 0 .
x = ...................., x = ...................., x = .................... [3]
(b) Sketch the graph of y = (x – 5)(2x2 – 18).
On your sketch, write the values where the graph crosses the x-axis and the y-axis.
y
O
x
[3]
© UCLES 2020
0580/PQ/20