Problem 20 - Algebra 2

Question

Use the graph of \(y=e^{x}\) to evaluate each expression to four decimal places. $$ e^{-2} $$

Step-by-Step Solution

Answer

The value of this expression \(e^{-2}\) is positive and it lies between 0 and 1, but for a precise decimal value a calculator is advisable. After performing the calculations, it is approximately 0.1353.

Step 1: Understanding the Properties of Exponential Function

First understand the basic property of the exponential function \(e^{x}\). It's always positive and it decreases as \(x\) decreases. However, it never reaches zero. So when \(x=-2\), we expect \(e^{-2}\) to be a positive small number.

Step 2: Navigating the Graph

Look at the graph of \(y=e^{x}\). Locate the \(x=-2\) on the x-axis. Trace a vertical line from \(x=-2\) upwards until it meets the curve of function \(y=e^{x}\). The y-coordinate of this point of intersection provides the value of \(e^{-2}\).

Step 3: Interpreting the result

Depending on the scale used in the graph, interpretation of the exact value will vary. The estimated value should lie between 0 and 1. Use a scientific calculator for more precise calculations.

Key Concepts

Graph of Exponential Functions

When dealing with exponential functions, it's crucial to understand their graphical representation. The graph of the exponential function, such as \(y = e^{x}\), showcases a smooth, continuously increasing curve. It passes through the point (0,1) and never touches the x-axis.

To understand this better, consider important characteristics:

• The function \(y=e^{x}\) rises sharply as \(x\) increases toward infinity; conversely, it approaches zero but never reaches it as \(x\) decreases toward negative infinity.
• It has a horizontal asymptote along the x-axis, meaning it gets closer and closer to zero but never actually becomes zero.

When working with the graph, you can visually determine values of the function for particular \(x\) values. For instance, to find \(e^{-2}\), locate \(x=-2\) on the x-axis and move up until you hit the curve, where the y-value is the answer.

Evaluating Expressions

Evaluating expressions involving exponential functions often involves calculating specific values that the function takes. In this context, you might use the graph, as mentioned in the previous section, or use a calculator for more precision.

Let's break down the key points:

• Direct evaluation: You can use a calculator capable of handling exponential expressions. For example, with \(x = -2\), you can directly compute \(e^{-2}\), which provides an accurate decimal approximation, often needed for tasks requiring precise numbers.
• Graph estimation: In scenarios without calculators, the graph helps to provide an estimate of \(e^{-2}\) by locating where \(x = -2\) intersects \(y = e^{x}\).

Calculators are handy when exact precision is required, usually delivering the result to four decimal places or beyond, such as approximately 0.1353 for \(e^{-2}\).

Properties of Exponential Functions

Exponential functions possess several intriguing properties that make them both elegant and powerful.

These functions, like \(y = e^{x}\), have distinct characteristics:

• They are continuous and differentiable over all real numbers, enabling smooth calculations and providing useful applications in calculus.
• Their rate of change is proportional to their current value; this property is the foundation for this function's applications in modeling growth and decay such as population growth, radioactive decay, and more.
• They are always positive functions, meaning no matter the input \(x\), the output \(e^x\) will always be positive.

Understanding these properties helps in anticipating how exponential functions behave under different scenarios. Recognizing these patterns allows users to apply these functions effectively in scientific and mathematical contexts.