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Magazine Chapter 4: Problem 99
For the following exercises, describe the end behavior of the graphs of the functions. $$f(x)=3(4)^{-x}+2$$
Short Answer
Expert verified
As x 鈫� 鈭�, f(x) 鈫� 2; as x 鈫� -鈭�, f(x) 鈫� 鈭�.
Step by step solution
01
Identify the Base Function
The base function in this case is an exponential function given by \(4^{-x}\). In exponential functions, the base indicates whether the graph will exhibit growth or decay. Here, the base is \(4\), and since it's in the form \(4^{-x}\), we have a decaying exponential function. This implies the function decreases as \(x\) increases.
02
Analyze Impact of Multiplication
The function is multiplied by \(3\), resulting in \(3(4)^{-x}\). Multiplying an exponential function by a constant stretches or compresses the graph vertically. Here, \(3\) causes a vertical stretch, but it does not affect the end behavior; it simply increases the rate of decay along the y-axis.
03
Study the Constant Addition
The function is shifted vertically by adding \(2\). Therefore, the function \(f(x)=3(4)^{-x}+2\) is shifted up by 2 units, which moves the horizontal asymptote from \(y=0\) to \(y=2\). This means as \(x\) approaches \( ext{-}\infty\), \(f(x)\) will approach \(2\).
04
Determine End Behavior as x Approaches Positive Infinity
As \(x\) approaches positive infinity, \(4^{-x}\) moves towards \(0\) because the term decreases rapidly to zero. The function hence approaches \(y o 2\) as \(x o \infty\).
05
Determine End Behavior as x Approaches Negative Infinity
As \(x\) approaches negative infinity, \(4^{-x}\) increases without bound, leading \(f(x)\) to increase. This implies the graph of \(f(x)\) rises sharply, moving toward positive infinity, so \(f(x) o \infty\) as \(x o ext{-}\infty\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Exponential Decay
Exponential decay occurs when a quantity decreases at a rate proportional to its current value. In exponential functions, such as the one given in the exercise, decay is represented by the negative exponent. When you see a base raised to a negative power, like in the function \(4^{-x}\), it indicates exponential decay.
Here's how it works:
Here's how it works:
- The base, \(4\), tells us the rate at which the exponential function decreases.
- The negative sign in the exponent \(-x\) indicates decay, meaning as \(x\) increases, the value of \(f(x)\) decreases.
- Exponential decay is characterized by a rapid reduction as you move along the positive axis.
Vertical Transformation
Vertical transformations involve shifting the graph of a function up or down. In the exercise, the exponential function \(3(4)^{-x}\) is transformed vertically by the addition of a constant.
- Adding a constant, in this case, \(+2\), means the entire graph moves up by two units.
- Vertical stretches or compressions can also occur when multiplying by a constant; here, multiplying by \(3\) stretches the graph vertically.
- These transformations don't change the fundamental shape but alter the position or steepness of the graph.
Horizontal Asymptote
A horizontal asymptote is a horizontal line that the graph of a function approaches but never actually reaches. In this problem, the original exponent function \(4^{-x}\) has an asymptote at \(y=0\).
- Adding \(+2\) to the function moves this horizontal asymptote from \(y=0\) to \(y=2\).
- As \(x\) moves towards infinity, the function \(3(4)^{-x}+2\) gets closer to \(y=2\), but never truly reaches it.
- Horizontal asymptotes provide a baseline for understanding how the function behaves at extreme values of \(x\).
Graph Behavior
The behavior of the graph of a function describes how it behaves as \(x\) moves towards infinity or negative infinity. Understanding this is crucial for grasping long-term trends.
- As \(x\rightarrow \infty\), the term \(4^{-x}\) shrinks, driving \(f(x)\) to approach the horizontal asymptote \(y=2\).
- As \(x\rightarrow -\infty\), \(4^{-x}\) increases, causing \(f(x)\) to rise rapidly towards positive infinity.
- The graph thus starts high from the left, descends as \(x\) nears 0, and levels out close to \(y=2\) towards the right.
