Problem 62 - Algebra 2

Question

What is the equation of the asymptote of \(y=15 \cdot\left(\frac{1}{3}\right)^{x} ?\) $$\begin{array}{llll}{\text { A. } y=1} & {\text { B. } y=0} & {\text { C. } y=x} & {\text { D. } y=\frac{1}{3}}\end{array}$$

Step-by-Step Solution

Answer

The equation of the asymptote of \(y=15 \cdot \left(\frac{1}{3}\right)^{x}\) is \(y=0\). Therefore, the correct answer is B.

Step 1: Understand the function

This function \(y=15 \cdot \left(\frac{1}{3}\right)^{x}\) represents an exponential decay, because the base (\(\frac{1}{3}\)) is less than 1. As \(x\) increases, the value of \(y\) decreases, getting closer and closer to a specific value but never reaching it. This specific value is the horizontal asymptote.

Step 2: Identify the horizontal asymptote

In the case of exponential decay, the function value y approaches 0 as x approaches positive infinity or negative infinity. Hence, the equation of the horizontal asymptote of this function is \(y=0\).

Key Concepts

Horizontal Asymptote

In the world of mathematics, a horizontal asymptote is a horizontal line that a graph approaches as the independent variable (usually represented by \(x\)) either heads towards positive or negative infinity. Imagine drawing a line on your graph that gets infinitely close but never quite touches the curve of your function. This line is called the horizontal asymptote. It's essential in determining the behavior of functions as \(x\) increases or decreases substantially.

For an exponential function, like our example \(y=15 \cdot \left(\frac{1}{3}\right)^{x}\), the horizontal asymptote can tell us the value that \(y\) approaches. When dealing with exponential decay, the value of \(y\) approaches 0 as \(x\) tends to infinity, which in this case gives us the horizontal asymptote of \(y=0\). Keep in mind:

• Horizontal asymptotes provide a boundary for the value of \(y\).
• They help visualize how functions behave as \(x\) moves towards large positive or negative values.

Recognizing and understanding the asymptotic behavior is crucial for analyzing functions and predicting their long-term behavior.

Exponential Function

An exponential function is a mathematical expression where a constant base is raised to a variable exponent, such as \(y=a \cdot b^{x}\). Here, \(a\) is the initial value (or coefficient), \(b\) is the base of the exponential function, and \(x\) is the exponent.

For our specific function \(y=15 \cdot \left(\frac{1}{3}\right)^{x}\), we have:

• \(15\) is the initial value. That is the starting point of our function, determining the initial amount or size before any exponential growth or decay takes place.
• The base is \(\frac{1}{3}\), which leads to exponential decay, because it is less than 1.

Exponential functions are often part of real-world phenomena, such as radioactive decay, population decline, or cooling of an object.

When dealing with exponential functions, remember:

• If the base \(b\) is greater than 1, the function exhibits exponential growth.
• If \(b\) is less than 1, it demonstrates exponential decay.

These functions are powerful tools for modeling various scenarios in science, finance, and much more!

Base Less Than One

When we discuss a base less than one in the context of exponential functions, we are delving into the heart of exponential decay. If a base such as \(b\) in an expression \(y=a \cdot b^{x}\) is less than 1, it means that as \(x\) increases, the value of \(y\) decreases. This is due to the fractional base diminishing the initial value progressively.

In our example, the base is \(\frac{1}{3}\). Here's what that means:

• Every time \(x\) increases by 1, the value of \(y\) becomes a third of what it was previously.
• This leads to a rapid decrease, showing a distinct characteristic of exponential decay.

The concept of a base less than one is a great tool for modeling situations where a quantity reduces over time.

Factors such as radioactive decay or population decline can vividly be represented with a base less than one, showing how values decrease steadily over time.
Understandably, it is essential not only to recognize this concept in equations but also to appreciate its practical implications in real-world scenarios.