Problem 6 - Precalculus Functions and Graphs

Question

Fill in the blank. The graph of \(f(x)=a^{x}\) has the \(x\) -axis as a(n) _________.

Step-by-Step Solution

Answer

The x-axis is a horizontal asymptote.

Step 1: Understand the nature of exponential functions

An exponential function of the form \(f(x) = a^x\) where \(a > 0\) and \(a eq 1\), rapidly increases or decreases as \(x\) moves away from zero. The graph of such functions only crosses the \(y\)-axis when \(f(x)\) is positive, in this case, it’s at \(y = 1\) because \(a^0 = 1\).

Step 2: Explore the behavior as \(x\) approaches negative infinity

As \(x \to -\infty\), the value of \(a^x\) approaches 0 but never actually reaches 0. The graph gets closer and closer to the \(x\)-axis but does not intersect or touch it.

Step 3: Identify the term for such behavior

A line that a graph approaches but never touches is known as an asymptote. Therefore, the \(x\)-axis acts as a horizontal asymptote for the exponential function \(f(x) = a^x\).

Key Concepts

Asymptote

An asymptote is a line that a graph approaches but never actually touches. This concept is key when studying different types of functions, especially exponential functions. Understanding asymptotes helps us frame how the function behaves as it moves towards very large or very small values. For example, for the function \(f(x) = a^x\), where \(a > 0\) and \(a eq 1\), the graph will get closer and closer to certain lines, called asymptotes, but will never intersect them. By knowing where these asymptotes are, we can more accurately predict how the graph of a function will look and behave.

Horizontal Asymptote

A horizontal asymptote is a specific type of asymptote that is parallel to the x-axis. For the function \(f(x) = a^x\), the x-axis is a horizontal asymptote. As \(x\) approaches negative infinity (\(x \to -\infty\)), the value of \(a^x\) gets closer and closer to 0. The graph will decrease towards the x-axis but never touch or cross it. This behavior is crucial in understanding exponential functions as it shows how the function behaves for very large negative values of \(x\). The horizontal asymptote essentially provides a 'boundary' that the function approaches but doesn't surpass.

Graph Behavior

The behavior of the graph of an exponential function like \(f(x) = a^x\) is very distinct and notable. Here are the key points to understand:
- **Rapid Increase or Decrease**: The function rapidly increases (for \(a > 1\)) or decreases (for \(0 < a < 1\)) as \(x\) moves away from zero.
- **Crossing the y-axis**: It only crosses the y-axis when \(y = 1\), because \(a^0 = 1\).
- **Approaching but Not Touching**: As \(x \to -\infty\), the graph approaches the x-axis, but will never touch or cross it. Hence, the x-axis is a horizontal asymptote. These principles help define the unique, exponential shape of the graph and allow us to predict how the function will behave across the entire range of x values.