Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, describe the end behavior of the graphs of the functions. \(f(x)=3\left(\frac{1}{2}\right)^{x}-2\)

Step-by-Step Solution

Answer

As \( x \to \infty \), \( f(x) \to -2 \); as \( x \to -\infty \), \( f(x) \to \infty \).

Step 1: Analyze the Base of the Exponential Function

The function is expressed as \( f(x) = 3\left(\frac{1}{2}\right)^{x} - 2 \). The base of the exponential part is \( \frac{1}{2} \), which is between 0 and 1. This tells us that as \( x \to \infty \), the power \( \left(\frac{1}{2}\right)^{x} \to 0 \).

Step 2: Consider the Vertical Shift

The function has a vertical shift of \(-2\), indicated by \(-2\) at the end of the function. This means the whole graph of the exponential function is shifted down by 2 units.

Step 3: Determine End Behavior as x Approaches Infinity

When \( x \to \infty \), \( \left(\frac{1}{2}\right)^{x} \to 0 \). So, \( f(x) = 3 \cdot 0 - 2 \to -2 \). Thus, the graph of the function approaches \( y = -2 \) from above.

Step 4: Determine End Behavior as x Approaches Negative Infinity

When \( x \to -\infty \), \( \left(\frac{1}{2}\right)^{x} \to \infty \). The function becomes \( f(x) = 3 \cdot \infty - 2 \to \infty \). As \( x \to -\infty \), the function value becomes very large.

Key Concepts

Understanding Exponential Functions

Exponential functions are a type of mathematical function involving a constant base raised to a variable exponent. These functions are expressed in the form \( f(x) = a \, b^x \), where \( a \) is a coefficient, \( b \) is a positive constant called the base, and \( x \) is the exponent. In the case of our function \( f(x) = 3\left(\frac{1}{2}\right)^{x}-2 \), the base \( b \) is \( \frac{1}{2} \). The behavior of the function heavily depends on the base value.

When the base of an exponential function is between 0 and 1, like \( \frac{1}{2} \), the function is a decreasing exponential function. As \( x \) increases, the value of \( b^x \) approaches zero because multiplying \( b \) by itself repeatedly makes it smaller. In our example, as \( x \to \infty \), \( \left(\frac{1}{2}\right)^x \to 0 \). Thus, the graph slopes downward as it moves right.

On the other hand, if the base were greater than 1, the function would exhibit a different behavior, increasingly growing as \( x \) increases. Understanding this difference is crucial for grasping the end behavior of exponential functions.

Impact of Graph Transformations

Graph transformations change the appearance of a graph without altering its fundamental nature. These include vertical shifts, horizontal shifts, reflections, stretches, and compressions. In our function \( f(x) = 3\left(\frac{1}{2}\right)^{x}-2 \), a key transformation is the vertical shift due to the \(-2\) term.

Vertical shifts move the graph up or down. Here, the \(-2\) means that every point on the graph of \( 3\left(\frac{1}{2}\right)^x \) is shifted 2 units downward. This affects the horizontal asymptote, moving it from \( y = 0 \), typical for functions without vertical shifts, to \( y = -2 \).

• Reflecting an exponential function across the x-axis involves multiplying the entire function by -1.
• A horizontal shift might involve adding or subtracting a number from \( x \) inside the exponent.
• A vertical stretch or compression would change the value of the coefficient \( a \).

Understanding these transformations helps us predict how the function's graph will look and behave. Being able to identify these shifts and transformations is essential for interpreting the graph correctly.

Exploring Asymptotic Behavior

Asymptotic behavior describes how a function behaves as it approaches a certain line, called an asymptote, but never actually reaches it. In the context of our function, the horizontal asymptote is affected by the transformations applied, specifically the vertical shift.

For \( f(x) = 3\left(\frac{1}{2}\right)^{x}-2 \), as \( x \to \infty \), the exponential term \( \left(\frac{1}{2}\right)^x \) approaches 0. Thus, the function approaches the line \( y = -2 \) from above, making \( y = -2 \) the horizontal asymptote. It means no matter how large \( x \) becomes, \( f(x) \) will never be less than \(-2\).

On the flip side, as \( x \to -\infty \), \( \left(\frac{1}{2}\right)^x \) actually results in a very large value, so \( f(x) \to \infty \), indicating there is no upper bound asymptote as \( x \to -\infty \). Recognizing and interpreting these behaviors helps inform both the direction of the graph and limitations in range.