Exponential Functions and their Graphs

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.

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.

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

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.

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 ) + (4 ) = 50.0 2.5 0.1 ◄ Evaluate 9(22.5) + (40.1) by keying in: 9 2 2.5 4 0.1 2.5 0.1 (cor. to 3 sig. fig.)

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.

For the graph of y = 2x, x 1 1 2 3 4 y 0.5 1 2 4 8 16 y 20 15 10 5 1 1 2 3 4 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.

For the graph of y = , x 4 3 2 1 1 y 16 8 4 2 1 0.5 1 y 16 8 4 2 1 0.5 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.

1 1 2 3 4 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.

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

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.

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.

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

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)

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

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.

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

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.