SUMMARY OF GRAPHS
Ellipses
x2 y2 (x – h)2 (y – k)2
+ =1 + =1
a2 b2 a2 b2
y y
1 b 3
b
k 2
a
0 x
-2 -1 0 1 2
–a a 1
0 x
-2 -1 0 1 2 3 4 5 6 7
-1
–b h
Hyperbolas
x2 y2 y 2 x2
– =1 – =1
a2 b2 b b2 a2
y y=ax y
3 3
b
2 2
y=ax
1 1
b
0 x 0 x
–a a
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-1 -1
–b b
-2 -2
y=–ax
-3
b -3
y=– x
a
b
The asymptotes are y = x.
a
(x – h)2 (y – k)2 (y – k)2 (x – h)2
– =1 – =1
a2 b2 b2 a2
y y
5 5
4 4
3 3
a b
k 2
k 2
1 1
0 x x
0
-3 -2 -1 0 1 2 3 4 5 6 7 8 9 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
-1
h -1
h
-2 -2
y–k x–h
b = a .
The asymptotes are found by solving
Graphing Techniques
ax + b If degree of numerator is 1 more the degree of the
y= x–c denominator, the graph has an oblique asymptote.
The vertical asymptote is x = c. x2 – 3x 4
E.g. y = x + 1 = x – 4 + x + 1 by long division.
The horizontal asymptote is y = a.
The oblique asymptote is y = x – 4.
y
5
4
y
4
3
y=a 3
2
2
1
0 x
-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
1 -1
-2
0 x
-3
-11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11
-4
-1
-5
-2 -6
-7
y=x–4
-3 -8
-9
-4
-10
-5 -11
x=c -12
x = –1
-13
-14
Transformation of Graphs
Translation parallel to y–axis: y = f(x) + a y
Translation parallel to x–axis: y = f(x + a) y
2 3
y = x2 + 1
1 2
y = (x – 1)2
0 x 1
-4 -3 -2 -1 0 1 2 3 4
2 2
y=x y = (x + 1)
0 x
-1
y=x –1 2 -4 -3 -2 -1 0 1 2 3 4
y = x2
-1
-2
Scaling parallel to y–axis: y = a f(x) y
Scaling parallel to x–axis: y = f(ax) y
3
2
2
y = 2 sin x
1
y = sin x
1
x 0 x
0
-4 -3 -2 -1 0 1 2 3 4
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
y = sin x
-1
-1
-2
y = sin 2x
-2
-3
Reflection along x–axis: y = – f(x) y
Reflection along y–axis: y = f(–x) y
2 2
x
y=e
1 1
y = ex y = e–x
0 x 0 x
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
-1
y = –ex -1
-2 -2
Sequence of Transformations:
shift left scale // x-axis scale // y-axis shift up
y = f(x) f(x + c) f(bx + c) a f(bx + c) a f(bx + c) + d
Curve Sketching
To draw y = | f(x) | and y = f ( | x | ):
y = sin x y
y = | sin x | y
3 3
2 2
1 1
0 x 0 x
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
-1 -1
-2 -2
-3 -3
y = sin | x | y
3
To draw y = f ( | x | ):
2
Remove the part of the curve y = f(x) on the left 1
x
of the y–axis. -7 -6 -5 -4 -3 -2 -1
0
0 1 2 3 4 5 6 7
Reflect the right hand side of the curve along the -1
y-axis. -2
-3
y = f(x)
1 y
To draw y = : (3, 4)
f(x) (0, 4)
1
(1) is undefined when f(x) = 0 vertical y=2
f(x)
asymptote. x
(2) Horizontal asymptote y = a becomes 1
1
horizontal asymptote y = .
a y=x x=2
(3) Vertical asymptotes become x-intercepts.
(4) Maximum point (a, f(a)) becomes 1
1 y = f(x)
minimum point (a, f(a) ), and vice versa. y
1
(5) The common points of y = f(x) & y = f(x)
y = 1/2 (3, 1/4)
are the points where y = 1. x
1 (0, 1/4) 2
(6) As f(x) 0, f(x) .
1
As f(x) , 0.
f(x) x=1
To draw y = f(x) : y = f(x) :
y
(1) f(x) is undefined when f(x) 1, then f(x) f(x).
(0, 2) (3, 2)
y= 2 x
To draw y = f(x), draw y = f(x) .
2
y=– 2 1
(0, –2) (3, –2)
x=2
To draw y = f (x)
y
x=2
(1) Vertical asymptotes remain the same.
(2) Horizontal asymptote y = a becomes y=1
horizontal asymptote y = 0. x
(3) Oblique asymptote y = ax + b becomes 0 3
horizontal asymptote y = a.
(4) Stationary point (a, f(a)) becomes x-
intercept (a, 0).
(5) If y = f(x) is increasing, then f '(x) > 0
(6) If y = f(x) is decreasing, then f '(x) < 0