1. Exponential Functions and their Graphs

2. For any value of x, there is only one corresponding value of y.
Consider y = 10x. x y 1 10 2
100 3
1000 … …
For any value of x, there is only one corresponding value of y.
y = 10x is a function of x.

3. where a > 0 and a  1, is called an exponential function
A function in the form
y = ax or f(x) = ax,
where a > 0 and a  1, is called
an exponential function
with base a.
For example:
y = 10x and
are exponential functions.

4. Since a > 0 and a  1, ax is defined for any real number x.
The domain of an exponential function is all real numbers.

5. How about the cases when a = 1 and a < 0?
This is a constant function but NOT an exponential function.
y = ax
= 1x
= 1
a< 0
ax is undefined for some values of x.
e.g is NOT a real number.

6. Follow-up question
Given that f (x) = 9(2x) and g(x) = 4x, find the value of f (2.5) + g(–0.1), correct to 3 significant figures.
f ( ) + g( )
= 9(2 ) + ( )
= 50.0
2.5
0.1
◄ Evaluate 9(22.5) + (40.1) by keying in:
2.5
0.1
(cor. to 3 sig. fig.)

7. How do the graphs of exponential functions look like?
You can plot the graphs of y = 2x and
y = and see how
they look like.

8. For the graph of
y = 2x, x 1 1 2 3 4 y y 20 15 10 5 
It always lies above the x-axis.
y = 2x
It has no maximum point, minimum point and axis of symmetry.
It cuts the y-axis at (0, 1). x
As x increases, it goes upwards from left to right.

9. For the graph of y = , x 4 3 2 1 1 y 16 8 4 2 1 0.51 y
4 3 2 1 1 x y 20 15 10 5
It has no maximum point, minimum point and axis of symmetry.
It cuts the
y-axis at (0, 1).
It always lies above the x-axis.
As x increases, it goes downwards from left to right.

10.  y 20 15 10 5 x
4 3 2 1 1 x y 20 15 10 5
y = 2x
In fact, the above graphs are typical graphs of y = ax for a > 1 and 0 < a < 1.

11. y y 20 15 10 5
4 3 2 1 1 20 15 10 5
y = ax
(a > 1)
y = 2x
y = ax
(0 < a < 1) x x 
In fact, the above graphs are typical graphs of y = ax for a > 1 and 0 < a < 1.

12. Common features for the graph of y = ax
2. The graphs never cut the x-axis. They lie above the x-axis.
(0, 1) x y
(0, 1) x y
1. The graphs cut the y-axis at (0, 1).
3. The graphs have neither a maximum point, a minimum point nor an axis of symmetry.

13. Differences for the graph of y = ax
The value of y increases as x increases.
The value of y decreases as x increases.
(0, 1) x y
(0, 1) x y
As x increases, the rate of increase of y becomes greater.
As x increases, the rate of decrease of y becomes smaller.
The value of y gets closer and closer to zero as x increases indefinitely.
The value of y gets closer and closer to zero as x decreases indefinitely.

14. The table below summarizes the features of the graphs of exponential functions for a > 1 and 0 < a < 1.

15. Have you noticed any relation between the graphs
of y = 2x and y = ?
The graph of
can be obtained by reflecting the graph of y = 2x about the y-axis, and vice versa.
Q’(2, 4)
Q(2, 4)
P’(1, 2)
P(1, 2)

16. In general, the graphs of y = ax and y =
show reflectional symmetry with each other about the y-axis.
axis of symmetry

17. Consider the following graphs.x y
y = 2x
y = 5x
y = 10x
y = 0.5x
y = 0.2x
y = 0.1x
The smaller the value of a, the flatter is the graph of y = ax.
0 < a < 1
The larger the value of a, the flatter is the graph of y = ax.

18. Follow-up question
In the figure, the graph of y = 4x cuts the y-axis at A.
(a) Write down the coordinates of A.
(b) Sketch the graph of . A x y
y = 4x

19. The graph of y = 4x cuts the y-axis at (0, 1).
(a) The coordinates of A are (0, 1).
(b) The graph of can be obtained by
reflecting the graph of y = 4x about the y-axis.