Exponential Functions
Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are commonly represented by the formula
\( y = a \times b^x \),
where \( a \) is the initial value, \( b \) is the base, a non-zero constant, and \( x \) is the exponent. In the context of the given exercise, the exponential function is
\( y = \(\frac{2}{5}\)^x \).
Here, \( \frac{2}{5} \) acts as the base and \( x \) is the exponent. This particular function does not have a coefficient multiplying the base, so in this case, \( a = 1 \).
Understanding the base is crucial in determining whether the function represents growth or decay. When the base is greater than 1, the function models exponential growth—meaning the values of \( y \) increase as \( x \) increases. Conversely, if the base is less than 1 but greater than 0, like \( \frac{2}{5} \) in our example, \( y \) decreases as \( x \) increases, leading to exponential decay.
Graphing Exponentials
Graphing an exponential function helps to visually represent the relationship between the variables \( x \) and \( y \). To effectively graph an exponential function, it's recommended to calculate the value of \( y \) for various \( x \) values, typically including negative values, zero, and positive values of \( x \).
For the function \( y = \(\frac{2}{5}\)^x \), choosing \( x \) values such as -2, -1, 0, 1, and 2 will create a range of \( y \) values to plot on a graph. As demonstrated in the solution, when \( x \) is negative, \( y \) becomes greater than 1 because the negative exponent signifies a reciprocal action.| Likewise, when \( x \) is 0, \( y \) equals 1, showing the starting point of the curve on the graph. As \( x \) becomes positive, \( y \) produces values less than 1, reflecting the decay nature of the function.
To graph these points, plot them on a coordinate plane and then draw a smooth curve that connects the dots, making sure the line continues to approach the \( x \)-axis but never actually touches it, indicative of the asymptotic nature of exponential decay functions.
Exponential Growth and Decay
Exponential growth and decay are two important concepts that describe how a quantity increases or decreases proportionally at a constant rate over time. In exponential growth, the quantity, such as population size or investment balance, increases rapidly over periods, with the growth rate dependent on the current value. This is represented mathematically by functions where the constant base is greater than 1.
In contrast, exponential decay describes a decreasing quantity, where the base of the exponential function is between 0 and 1. Phenomena such as radioactive decay, cooling of substances, and depreciation of assets are examples where exponential decay provides a good model. With decay, the quantity decreases by a proportion that becomes smaller over time.
The function \( y = \(\frac{2}{5}\)^x \) in the original exercise is an example of exponential decay. As the base \( \frac{2}{5} \) is less than 1, the resultant \( y \) values get smaller as \( x \) increases, indicating a consistent rate of decay. This understanding is crucial in fields like finance, biology, and physics where analyzing growth and decay can yield insights into future trends and behaviors.
