Problem 97 - Precalculus

Question

For the following exercises, describe the end behavior of the graphs of the functions. $$f(x)=-5(4)^{x}-1$$

Step-by-Step Solution

Answer

As \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to -1 \).

Step 1: Identify the Function Type

The given function is an exponential function of the form \( f(x) = a \cdot b^{x} + c \), where \( a = -5 \), \( b = 4 \), and \( c = -1 \). Exponential functions are characterized by constants raised to the power of the variable \( x \).

Step 2: Analyze the Base of the Exponential Function

The base \( b = 4 \) is greater than 1, indicating exponential growth if \( a \) were positive. However, since \( a = -5 \), the direction of growth is flipped, so the function actually reflects exponential decay.

Step 3: Evaluate the End Behavior as \( x \to \\infty \)

As \( x \to \infty \), the term \( (4)^x \) grows very large. Because of the negative sign from \( a = -5 \), the term \( -5(4)^x \) will approach \(-\infty\). Adding \(-1\) causes \( f(x) \to -\infty \).

Step 4: Evaluate the End Behavior as \( x \to -\\infty \)

As \( x \to -\infty \), the term \( (4)^x \) approaches 0. Therefore, \( -5(4)^x \) approaches 0. With the additional \(-1\), the function approaches \(-1\).

Step 5: Summarize the End Behavior

The end behavior of the function \( f(x) = -5(4)^x - 1 \) is as follows: as \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to -1 \).

Key Concepts

Exponential Functions

Exponential functions are widely used in various fields such as biology, finance, and physics because they can effectively model growth and decay processes. These functions are generally of the form \( f(x) = a \cdot b^{x} + c \), where:

• \( a \) is a coefficient that affects the vertical stretch or compression of the graph.
• \( b \) is the base of the exponential expression and it determines the growth or decay rate.
• \( c \) is a constant that shifts the function vertically.

The function in our exercise is \( f(x) = -5(4)^{x} - 1 \), which is a quintessential exponential function. Here, the base \( b = 4 \) indicates growth, but the negative coefficient \( a = -5 \) flips the growth downward, modifying the function’s end behavior. Exponential functions exhibit rapid changes in value; therefore, understanding their nature is crucial in identifying how functions behave at their extremes.

End Behavior Analysis

End behavior refers to how the values of a function behave as the input \( x \) moves towards positive or negative infinity. It helps in understanding the general direction in which a graph moves.In the given function \( f(x) = -5(4)^x - 1 \):

• For \( x \to \infty \), \( (4)^x \) increases significantly. Multiplying by a negative \( -5 \) results in \( -5(4)^x \) approaching \(-\infty\). Thus, \( f(x) \to -\infty \) as \( x \to \infty \).
• For \( x \to -\infty \), \( (4)^x \) becomes a very small positive fraction, approaching 0. Hence, \( -5(4)^x \to 0 \). But due to the constant term \(-1\), the function approaches \(-1\). Hence, \( f(x) \to -1 \) as \( x \to -\infty \).

This means that the function rapidly declines as it moves towards the right on the graph, but levels off as it moves left, approaching \(-1\). This visual understanding is very useful when sketching or analyzing graphs for real-world contexts.

Function Transformations

Function transformations involve altering a function to shift, stretch, compress, or reflect its graph. They can drastically change the appearance or position of the graph, while the basic shape tied to the function type often remains recognizable.For the function \( f(x) = -5(4)^x - 1 \), we observe the following transformations:

• The negative coefficient \( a = -5 \) causes a reflection across the x-axis, flipping the function upside down.
• The value \( 4 \) is the base that influences the rate at which the graph either grows or decays, depending on the sign of \( a \).
• The subtraction by 1 shifts the entire graph downward by one unit, lowering its horizontal asymptote.

These transformations show how exponential functions can be manipulated to fit different modeling scenarios, adapting to various data sets and real-life situations.

Exponential Decay

Exponential decay describes a process where the quantity decreases rapidly at first and then slowly over time. Even though our function’s base \( b = 4 \) suggests growth, the negative sign before the coefficient \( a = -5 \) inverts this process, functioning like decay.In the function \( f(x) = -5(4)^x - 1 \):

• The negative coefficient results in the exponential decay pattern, causing the function to decrease as \( x \) increases.
• Contrary to typical decay where \( 0 < b < 1 \), here the large positive base results in rapid decline because of the negative sign in front.

This decay is crucial in contexts where something is reducing in amount or intensity, such as the cooling of a hot object, radioactive decay or investment depletion. Understanding exponential decay enables predictions about when a process will slow down or approach zero.