Question

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Answer 1

Answer:

Definitions

At the most basic level, an exponential function is a function in which the variable appears in the exponent. The most basic exponential function is a function of the form y=bx where b is a positive number.

When b>1 the function grows in a manner that is proportional to its original value. This is called exponential growth.

When 0>b>1 the function decays in a manner that is proportional to its original value. This is called exponential decay.

Graphing an Exponential Function

Example 1

Let us consider the function y=2x when b>1. One way to graph this function is to choose values for x and substitute these into the equation to generate values for y. Doing so we may obtain the following points:

(−2,14), (−1,12), (0,1), (1,2) and (2,4)

As you connect the points, you will notice a smooth curve that crosses the y-axis at the point (0,1) and is increasing as x takes on larger and larger values. That is, the curve approaches infinity as x approaches infinity. As x takes on smaller and smaller values the curve gets closer and closer to the x-axis. That is, the curve approaches zero as x approaches negative infinity making the x-axis is a horizontal asymptote of the function. The point (1,b) is on the graph. This is true of the graph of all exponential functions of the form y=bx for x>1.

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