1. Graphs of
Related
Functions

2. The graphs of all functions cut the x-axis where y = 0
Points to Remember
The graphs of all functions cut the x-axis where y = 0
The graphs of all functions cut the y-axis where x = 0

3. Graph of –f(x) Transformations reflects f(x) in the x - axis
y = f(x)
y = x2
y = – f(x)
y = –x2
y = –f(x)
reflects f(x) in the x - axis

4. reflects f(x) in the x - axis
y = f(x)
y = x3
y = – f(x)
y = – x3
y = –f(x)
reflects f(x) in the x - axis

5. Graph of f(– x) Transformations reflects f(x) in the y - axis
y = f(x)
y = (x – 2)2
Happy parabola
x = 0  y = (–2)2 = 4
y = 0  (x – 2)2 = 0
 x = 2 (twice)
So x-axis tangent
y = f(– x )
y = (– x – 2)2
y = f(– x)
reflects f(x) in the y - axis

6. Graph of f(x) ± k Transformations Depending on the value of k
y = f(x)
y = x2
y = f(x) ± k
y = x2 + 3
y = x2 – 1
y = f(x) ± k
moves f(x) up or down
Depending on the value of k
+ k  move up
– k  move down

7. Graph of f(x ± k) Transformations
y = f(x)
y = f(x ± k)
y = x2
y = (x – 1)2
y = (x + 2)2
y = f(x ± k)
moves f(x) to the left or right depending on the value of k
– k  move right
+ k  move left

8. Graph of k f(x) Transformations
y = f(x)
y = x2 – 1
y = k f(x)
y = 4(x2 – 1)
y = 0.25(x2 – 1 )
y = k f(x)
Multiply y coordinate by a factor of k
k > 1  (stretch in y-axis direction)
0 < k < 1  (squash in y-axis direction)

9. Graph of f(kx) Transformations
y = f(x)
y = f(2x)
(- 2 , 10)
- 3
- 6
( - 4, 10)
y = f(x)
( - 4, 10)
(1,8)
(2,8)
f(kx) : The value of k scales the graph in x-direction
k = 2, x - scale factor = 1/2

10. Graph of f(kx) Transformations
(1,13)
(3,8) 8
y = f(x)
y = f(1/2x)
y = f(x)
(3,8)
(1,13) 8
(2,13)
(6,8)
f(kx) : The value of k scales the graph in x-direction
k = 1/2, x - scale factor = 2

11. Graph moves 2 units to the right
y = logax
(a , 1)
(a + 2 , 1)
(1 , 0)
(3 , 0)
y = loga(x – 2)
Graph moves 2 units to the right

12. Find the values of a and b.
y = loga(x + b)
Find the values of a and b.
( 22 , 2)
3 units left  b = 3
( –2 , 0)
( 1 , 0)
y = loga(x + 3)
When x = 22, y = 2
2 = loga(22 + 3)
y = log5(x + 3)
2 = loga 25  a = 5

13. y = abx
Find the values of a and b.
( 2 , 48)
y = abx
When x = 0, y = 3
becomes y = 3bx
3 = a×b0
When x = 2, y = 48
3 = a×1
48 = 3b2
a = 3
16 = b2
( 0 , 3)
b = 4
y = 3 × 4 x

14. The following questions are on
Graphs & Functons
Non-calculator questions will be indicated

15. Reflect across the y axis Now scale by 2 in the y direction
The diagram shows the graph of a function f.
f has a minimum turning point at (0, -3) and a point of inflexion at (-4, 2).
a) sketch the graph of y = f(-x).
On the same diagram, sketch the graph of y = 2f(-x) a)
Reflect across the y axis b)
Now scale by 2 in the y direction

16. Determine the values of a, b and c
The diagram shows a sketch of part of the graph of a trigonometric function
Determine the values of a, b and c 2a 1
1 in 
2 in 2 
a is the amplitude:
a = 4
b is the number of waves in 2
b = 2
c is where the wave is centred vertically
c = 1

17. 10 - f(x) is positive for -1 < x < 5
a) Express f(x) = x2 – 4x in the form (x – a)2 + b
b) On the same diagram sketch
i) the graph of y = f(x)
ii) the graph of y = 10 - f(x)
c) Find the range of values of x for
which 10 - f(x) is positive
(2, 9) b) a) c)
Solve:
10 - f(x) is positive for < x < 5

18. Sketch the graph of the derived function f ’.
The graph of a function f intersects the x-axis at (–a, 0) and (e, 0) as shown. There is a point of inflexion at (0, b) and a maximum turning point at (c, d).
Sketch the graph of the derived function f ’.
m is +
m is +
m is -
f(x)

19. The diagram shows the graphs of two quadratic functions y = f(x) and y = g(x)
Both graphs have a minimum turning point at (3, 2). Sketch the graph of y = f ’(x) and on the same diagram sketch the graph of y = g‘(x)
y=f(x)
y=g(x)

20. Part of the graph of y = f(x) is shown in the diagram.
On separate diagrams sketch the graph of
a) y = f(x+1) b) y = – 2f(x)
Indicate on each graph the
images of O, A, B, C, and D. a)
graph moves to the left 1 unit b)
graph is reflected in the x axis
graph is then scaled 2 units in the y direction

21. (1,4)
(6,-8) 3 7
y = – 2f(x)
A(-1,2) 3 7
y = f(x)
C(6,4)
C(5,4)
y = f(x+1) 7 2 -1 -2