Problem 29 - College Algebra

Question

Describe the end behavior of the graphs of the functions. $$f(x)=-5(4)^{x}-1$$

Step-by-Step Solution

Answer

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

Step 1: Identify the Function Type

The given function, \(f(x) = -5(4)^x - 1\), is an exponential function. Exponential functions are characterized by having a constant base raised to a variable exponent.

Step 2: Understand Base Effects

Here, the base is 4, which is greater than 1. In an exponential function \(a \cdot b^x\), when \(b > 1\), \(b^x\) grows as \(x\) increases. However, in this function, \(b^x\) is multiplied by a negative constant (-5), which affects its direction of growth.

Step 3: Analyze Horizontal Asymptote

As \(x\) approaches infinity, the term \(-5(4)^x\) dominates the function, causing \(f(x)\) to decrease towards negative infinity since the base is greater than 1. The horizontal asymptote is determined by the constant term, \(-1\), affecting the entire function but not the strong influence of the exponential term as \(x\) grows large.

Step 4: Determine End Behavior as \(x \to \infty\)

When \(x\) approaches positive infinity, \(f(x)\) will approach negative infinity due to the negative coefficient \(-5\) multiplying the exponentially increasing term \((4)^x\). This means the function will decrease without bound as \(x\) increases.

Step 5: Determine End Behavior as \(x \to - fty\)

As \(x\) approaches negative infinity, \((4)^x\) becomes very small, approaching zero, since any positive number raised to a negative power results in a small positive fraction. Thus, \(f(x) = -5(4)^x - 1\) will approach the horizontal asymptote \(y = -1\).

Key Concepts

End Behavior

In mathematics, the end behavior of a function describes how the function behaves as the input variable, \(x\), approaches infinity or negative infinity. This concept helps us understand the long-term tendencies of the function's graph.

• For the function \(f(x) = -5(4)^x - 1\), as \(x\) approaches positive infinity \((x \to \infty)\), the term \((4)^x\) increases rapidly because 4 is greater than 1. When multiplied by \(-5\), a negative number, the overall expression \(-5(4)^x\) becomes increasingly negative.
• Thus, the function \(f(x)\) decreases towards negative infinity.
• Conversely, as \(x\) approaches negative infinity \((x \to -\infty)\), the term \((4)^x\) becomes smaller and closer to zero because 4 raised to a negative power results in a small fraction.
• This makes the entire function \(f(x)\) approach \(-1\), which is the constant term of the function.

Understanding these trends in end behavior gives insights into the general position and direction of the graph of the function over the entire \(x\)-axis.

Horizontal Asymptote

A horizontal asymptote is a horizontal line that the graph of a function approaches but never touches as \(x\) approaches infinity or negative infinity. For many exponential functions, horizontal asymptotes signify a limiting behavior.

• The horizontal asymptote for the function \(f(x) = -5(4)^x - 1\) is \(y = -1\).
• This is determined by considering the effect of the constant term when \((4)^x\) becomes negligible as \(x\) approaches negative infinity.
• As the exponent becomes large and negative, the term \(4^x\) becomes very close to zero, making \(-5(4)^x\) tend towards zero.
• Thus, the value of \(f(x)\) approaches \(-1\), effectively forming a horizontal line that the function approaches but does not reach.

Understanding the concept of horizontal asymptotes helps in visualizing where the function 'settles' as it moves further along the \(x\)-axis.

Negative Exponent

A negative exponent indicates the reciprocal of the base raised to the corresponding positive power. This property drastically affects the value of expressions within a function, especially as \(x\) approaches negative infinity.

• In the function \(f(x) = -5(4)^x - 1\), when \(4^x\) has a negative exponent, it means \(4^x\) is the same as \(\frac{1}{4^{|x|}}\).
• This transformation causes \(4^x\) to become very small, approaching zero, as \(x\) takes on negative values.
• The negative exponent effectively "shrinks" the base, decreasing its impact on the whole function.
• For exponential functions, this behavior creates a diminishing impact from exponential growth, showcasing contrasting behavior when compared to positive exponents.

Grasping the effect of negative exponents ensures a deepened understanding of how exponential functions 'decay' or diminish as \(x\) becomes less potent.

Function Graph

Understanding the graph of a function, especially an exponential function like \(f(x) = -5(4)^x - 1\), is key to visualizing mathematical concepts.

• The graph of this function generally showcases a steep decline as \(x\) increases due to the negative coefficient \(-5\) multiplying the exponentially growing term \((4)^x\).
• As \(x\) approaches positive infinity, the graph extends downward, behaving asymptotically towards negative infinity.
• Conversely, towards negative infinity, the graph flattens and aligns itself closer to the line \(y = -1\), the horizontal asymptote.
• This results in a graph that is continually decreasing but never reaching the horizontal line at \(y = -1\).

Visually, the graph of an exponential function like this helps in understanding complex concepts like growth, decay, and limit behavior as expressed through horizontal asymptotes and end behavior.