Exponential functions A function in which x is in the power is called an exponential function. Exponent is another name for power. The general form is: y = b(ax) where a and b are constants and a is positive. The graph of an exponential function is called an exponential curve. An asymptote is a line that a curve approaches, but never touches.

Example 1 y = 10x y = 3x y = 2x Sketch the following curves y = 2x x -3 -2 -1 1 2 3 y 1/8 ¼ ½ 1 2 4 8 y = 3x x -3 -2 -1 1 2 3 y 1 3 9 27 All curves of the form ax pass through the point (0,1) 0·037 0·11 0·33 Also sketched is 10x and 1·4x. As the value of a gets bigger the curve gets: steeper to the right of the y-axis and closer to x-axis to the left of the y-axis. The x-axis is an asymptote. y = 1·4x

Example 2 y = 3(2x) Sketch the following curves y = 2x y = 0·3(2x) x -3 -2 -1 1 2 3 y 0·04 0·08 0·15 0·3 0·6 1·2 2·4 y = 3(2x) x -3 -2 -1 1 2 3 y 3/8 ¾ 1½ 3 6 12 24 Also sketched is 2x. As the value of b gets bigger in y = b(ax): the curve gets steeper The curve cuts the y-axis at b instead of 1. y = 0·3(2x)

Example 3 Today’s work y = 1·2(2x) x -3 -2 -1 1 2 3 y -0·3 -0·6 -1·2 1 2 3 y -0·3 -0·6 -1·2 -2·4 -4·8 As b is negative the curve is upside down As b is 1·2 the curve cuts the y-axis at 1·2 -0·15 -9·6 Today’s work Exercise 12 D page 371 #6, 9, 10, 11