Question

In Exercises 15–22, tell whether the function represents exponential growth or exponential decay. Then graph the function. $$ y=e^{-2 x} $$

Short answer

The function \(y=e^{-2x}\) represents an exponential decay. Its graph is a curve that starts at the y-intercept (0,1) and slopes downwards, approaching but never reaching the x-axis.

Step-by-step solution

Identify Nature of the Exponential Function

The function given is \(y=e^{-2x}\). ‘e’ is the base of the function which is the mathematical constant approximately equal to 2.71828, and always positive. The exponent of the function is -2x, which is negative. Therefore, this indicates that the function represents exponential decay.

Determine Key Points for the Graph

To graph the function, we need to find key points. Choose a few values for x and calculate the corresponding y-values. For example, when \(x=0\), \(y=e^{-2*0}=e^{0}=1\), and when \(x=1\), \(y=e^{-2*1}=e^{-2}≈0.135\). So two points on the graph are (0,1) and (1, 0.135).

Plot the Function

Using these points, along with several others, we can draw the graph of the function. The y-intercept is at (0,1). As \(x\) moves towards infinity, the \(y\) values approach 0, showing an exponential decay. The curve of decay becomes closer to the x-axis but never crosses it, and so the x-axis is a horizontal asymptote for the function.

Key concepts

Graphing Functions

Visualizing an equation often simplifies understanding, especially when dealing with abstract concepts like functions. Here, the function given is an exponential function: \(y=e^{-2x}\). When graphing such functions, the key steps include:

• Identifying critical points such as intercepts: For \(y=e^{-2x}\), this would include the y-intercept where \(x=0\).
• Determining the shape of the curve: Exponential decay results in a curve that slopes downward.
• Analyzing behavior as \(x\) becomes very large or very small.

For this particular graph, the y-intercept is at (0, 1), indicating that when \(x=0\), \(y\) is 1. As \(x\) increases, the function's value decreases gradually, evidencing the nature of exponential decay. It's essential to plot several points to accurately draw the continuous curve that never touches the x-axis.

Exponential Functions

Exponential functions are a cornerstone in mathematics, described by the general form \(y=a\cdot b^{x}\). In these functions, 'a' influences the starting point, and 'b' determines the direction (growth or decay) and rate of change.

• If \(b>1\), it signifies exponential growth.
• If \(0

Exponential decay, such as in the function \(y=e^{-2x}\), happens when the base of the exponent is a fraction or a number less than 1, often due to a negative exponent. This results in the function value decreasing as \(x\) increases, illustrating how rapidly an amount can shrink over time.

Mathematical Constants

Mathematical constants are unique numbers found consistently in mathematical formulas and functions. The number \(e\) is one such constant, essential in natural processes that display continuous growth or decay.

• \(e\) is known as Euler's number, approximately equal to 2.71828.
• It serves as the base for natural logarithms.

In our function \(y=e^{-2x}\), \(e\) acts as the base of the exponential, which adjusts the rate of decay. The constant's properties make it ideal for modeling continuous processes, such as population decline or radioactive decay, providing insight and approximations in real-world applications.

Horizontal Asymptotes

A horizontal asymptote in graphing signifies a line that the curve approaches but never actually touches. This behavior is a distinct feature of exponential functions, especially in decay scenarios.

• In the function \(y=e^{-2x}\), the x-axis \(y=0\) serves as a horizontal asymptote.
• The curve comes infinitely close to this line as \(x\) increases without ever crossing it.

This characteristic arises because the exponential term shrinks towards zero as \(x\) grows. Knowing about horizontal asymptotes helps in predicting long-term behavior of functions and understanding limits, crucial for studies involving calculus and analytic geometry.