![]() ![]() Make sure you understand the definitions.Īn exponent is a numeral used to indicate how many times a factor is to be used in a product. When naming terms or factors, it is necessary to regard the entire expression.įrom now on through all algebra you will be using the words term and factor. ![]() Rules that apply to terms will not, in general, apply to factors. It is very important to be able to distinguish between terms and factors. When an algebraic expression is composed of parts to be multiplied, these parts are called the factors of the expression. In 2x + 5y - 3 the terms are 2x, 5y, and -3. When an algebraic expression is composed of parts connected by + or - signs, these parts, along with their signs, are called the terms of the expression. Since these definitions take on new importance in this chapter, we will repeat them. In other words, the rate of change of the graph of e x is equal to the value of the graph at that point.In section 3 of chapter 1 there are several very important definitions, which we have used many times. One important property of the natural exponential function is that the slope the line tangent to the graph of e x at any given point is equal to its value at that point. Like the exponential functions shown above for positive b values, e x increases rapidly as x increases, crosses the y-axis at (0, 1), never crosses the x-axis, and approaches 0 as x approaches negative infinity. Since any exponential function can be written in the form of e x such thatĮ x is sometimes simply referred to as the exponential function. The natural exponential function is f(x) = e x. Compared to the shape of the graph for b values > 1, the shape of the graph above is a reflection across the y-axis, making it a decreasing function as x approaches infinity rather than an increasing one. when 0 0 in that the function is always greater than 0, crosses the y axis at (0, 1), and is equal to b at x = 1 (in the graph above (1, ⅓)). In the table above, we can see that while the y value for x = 1 in the functions 3x (linear) and 3 x (exponential) are both equal to 3, by x = 5, the y value for the exponential function is already 243, while that for the linear function is only 15. Just as an example, the table below compares the growth of a linear function to that of an exponential one. The key characteristic of an exponential function is how rapidly it grows (or decays). ![]() The graph above demonstrates the characteristics of an exponential function an exponential function always crosses the y axis at (0, 1), and passes through a (in this case 3), at x = 1. Below is the graph of the exponential function f(x) = 3 x. ![]() There is a horizontal asymptote at y = 0, meaning that the graph never touches or crosses the x-axis. For negative x values, the graph of f(x) approaches 0, but never reaches 0. when b > 1įor f(x) = b x, when b > 1, the graph of the exponential function increases rapidly towards infinity for positive x values. This is because 1 raised to any power is still equal to 1. When b = 1 the graph of the function f(x) = 1 x is just a horizontal line at y = 1. There are a few different cases of the exponential function. The rate of growth of an exponential function is directly proportional to the value of the function. Exponential functionĪn exponential function is a function that grows or decays at a rate that is proportional to its current value. It may also be used to refer to a function that exhibits exponential growth or exponential decay, among other things. In algebra, the term "exponential" usually refers to an exponential function. Home / algebra / exponent / exponential Exponential ![]()
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