![]() The limit of the function as x approaches positive infinity is ∞. If the base of the exponential function, b, is greater than 1, the output values of the function will increase without bound as x approaches positive infinity. If we add a constant k to the function f(x) the graph will be shifted k units up or down but the shape of the graph will not change, this means that if the graph of g(x) = f(x) + k is an exponential function, then f(x) is also an exponential function.įor an exponential function in general form, as the input values increase without bound, the output values will increase without bound or will get arbitrarily close to zero. In other words, the graph of an exponential function is always increasing or always decreasing, and it's concave up if the base is greater than 1 or concave down if the base is between 0 and 1. This is, the ratio of the y-coordinates of two points on the graph of the function is the same for any two points. This can be proven by observing that if the values of g(x) are proportional over equal length input-value intervals, then the graph of g(x) is exponential, and since g(x) = f(x) + k, the graph of f(x) is also exponential.Įxponential functions are functions that change proportionally over equal-length input-value intervals. ![]() If the values of the additive transformation function g(x) = f(x) + k of any function f are proportional over equal length input-value intervals, then f is exponential. The additive transformation function g(x) shifts the graph of f(x) vertically by k units. Source: Statistics How To Additive TransformationsĪn additive transformation of a function f(x) is a function of the form g(x) = f(x) + k, where k is a constant. If the base, b, is between 0 and 1, the exponential function demonstrates exponential decay, and its graph is concave down. ![]() If the base, b, is greater than 1, the exponential function demonstrates exponential growth, and its graph is concave up. Downīecause the output values of exponential functions are proportional over equal-length input-value intervals, the graphs of exponential functions are always increasing or always decreasing.ĭepending on the value of the base, b, the function will have an upward or downward concavity. Here the base 1.05 represents the growth rate of 5% and the exponent represents the number of years. For example, if a population is growing at a rate of 5% per year, the population after n years will be P ∗ 1.0 5 n P*1.05^n P ∗ 1.0 5 n, where P is the initial population. This idea can be used to model real-world situations that involve exponential growth or decay, such as compound interest, population growth, and radioactive decay. It's important to note that in both cases the initial value a must be greater than 0, otherwise the function would not be defined. The smaller the base, the faster the decay. This means that as x increases, the value of the function, f(x), decreases at an increasingly rapid rate. On the other hand, when the base, b, is between 0 and 1 ( 0 < b < 1), the exponential function demonstrates exponential decay. The larger the base, the faster the growth. This means that as x increases, the value of the function, f(x), increases at an increasingly rapid rate. When the base, b, is greater than 1 ( b > 1), the exponential function demonstrates exponential growth. ![]() The behavior of an exponential function depends on the value of the base, b. The general form of an exponential function is f ( x ) = a b x f(x) = ab^x f ( x ) = a b x, where a is the initial value, also known as the y-intercept, and b is the base, which is a positive number other than 1. ![]() An exponential function is a function in which the variable, x, appears in the exponent, rather than in the base. ![]()
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