The doubling function, for example, is a power function.
Use the rule: y = 2x. Use values of 1, 2, 3, and 1/2 for x and get corresponding values of 2, 4, 8, and 2 for y values.
Examine the "Family of Exponentials Graphs" to see how different y = 1x behaves relative to y = 1.5x, y = 2x, y = ex. For every value of x, 1x is 1, so the function is constant, there is no growth.
For all bases greater than 1, there is ALWAYS GROWTH, as x increases.
For all bases between 0 and 1, there is ALWAYS DECAY as x increases.
For positive values of x, the larger the base, 1.5 versus 2, versus e, the larger the value of the function, y = 1.5x, y = 2x, y = ex, and therefore the steeper the curve, the faster the growth: The larger the base, the faster the growth.
For negative values of x, the smaller the base, 1.5 versus 2, versus e, the larger the value of the function, y = 1.5x, y = 2x, y = ex, the and the steeper the curve, and the faster the growth: For negative values of x, the smaller the base, the faster the growth.
The slope of y = bx is bx(ln b).
The inverse function for y = bx is y = logb x.
Power or exponential functions are powerful tools. They are used to express growth and decay. They are used to express other function such as:
The numbers 0, 1, -1, , e, and i are celebrates among numbers. They are no different than other more common numbers, but are "more famous" and very useful and important. The numbers , 0, 1, and -1 are probably familiar to you. The complex number i is the square root of negative one, -1. It solves the equation x² = -1 with solutions of i and -i.
The number e is about equal to 2.71828 but this does not explain what number it is. It also may be said to solve an equation. One definition of e involves the solution of a calculus equation involving an integration, an antiderivative, the inverse of finding the slope. Here's the equation:
Consider the graph of the horizontal line y = 0 as a bottom boundary line. Use the vertical line x = 1 as a left boundary line, and the reciprocal function as a top boundary line. A four-sided shape would be created if a vertical line were drawn on the right. The vertical line which creates a shape having an area of one is the vertical line x = e. Suppose the integral of the reciprocal from 1 to some upper limit equals one. That upper limit is e. The number e solves the equation below where the upper limit b is the value being sought.
You may experiment with this yourself. Find the upper limit, the value that
makes the equation equal to 1.
© 2005, Agnes Azzolino