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Magazine Chapter 1: Problem 31
Graph the function \(y = x ^ { n } 2 ^ { - x } , x \geq 0 ,\) for \(n = 1,2,3,4,5\) and \(6 .\) How does the graph change as n increases?
Short Answer
Expert verified
As \(n\) increases, the function's peak grows taller and shifts closer to \(x=0\) with sharper initial increases.
Step by step solution
01
Understanding the Function
The function we are dealing with is \(y = x^n 2^{-x}\), where \(n\) is a parameter taking values 1,2,3,4,5, and 6. The term \(x^n\) is a polynomial component which increases as \(n\) increases, and the term \(2^{-x}\) is an exponential decay factor which decreases as \(x\) increases.
02
Set Up for Graphing
We will graph the function for each value of \(n\) separately: 1, 2, 3, 4, 5, and 6. For each graph, observe the behavior of the function as \(x\) increases starting from \(x = 0\).
03
Graphing for n=1
For \(n=1\), the function is \(y = x \cdot 2^{-x}\). As \(x\) increases, initially the function increases because \(x\) is growing faster than \(2^{-x}\) is decaying, but eventually the decay of \(2^{-x}\) dominates.
04
Graphing for n=2
For \(n=2\), the function is \(y = x^2 \cdot 2^{-x}\). Compared to \(n=1\), the function will rise faster initially due to the quadratic term \(x^2\), but the decay from \(2^{-x}\) will still cause it to eventually decrease.
05
Graphing for n=3
For \(n=3\), the function is \(y = x^3 \cdot 2^{-x}\). The initial rise in the graph is steeper compared to \(n=2\) because the cubic term \(x^3\) allows the function to grow faster before being dampened by \(2^{-x}\).
06
Graphing for n=4
For \(n=4\), the function is \(y = x^4 \cdot 2^{-x}\). The rise is even more pronounced as the polynomial term increases, showing a stronger initial growth but once again dropping due to the exponential decay of \(2^{-x}\).
07
Graphing for n=5 and n=6
For \(n=5\) and \(n=6\), the functions \(y = x^5 \cdot 2^{-x}\) and \(y = x^6 \cdot 2^{-x}\) continue this trend of a sharper initial rise with a subsequent steep decline. The higher the power in \(x^n\), the larger the peak becomes before the function decreases.
08
Analyzing Graph Changes
As \(n\) increases, the peak of the graph occurs at a higher value of \(y\) and closer to \(x = 0\). The function initially grows faster and reaches a higher maximum before declining, reflecting more pronounced peaks.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Functions
Polynomial functions form a crucial part of the function given, which is structured as \( y = x^n \cdot 2^{-x} \). Here, \( x^n \) represents the polynomial component of the equation. As you increase the value of \( n \) from 1 to 6, you are essentially changing the degree of the polynomial. The degree of a polynomial, denoted as the highest power of \( x \), significantly impacts the behavior of the function.
Key characteristics of polynomial functions include:
Key characteristics of polynomial functions include:
- **Degree Impacts Growth Rate**: Higher degrees mean faster growth rates. For example, a cubic polynomial \( x^3 \) grows faster than a quadratic \( x^2 \).
- **Shape and Extent**: With increasing \( n \), the polynomial component contributes to a sharper and more distinct curve in the graph as the function initially rises.
Exponential Decay
In the function \( y = x^n \cdot 2^{-x} \), the term \( 2^{-x} \) represents exponential decay. Exponential decay occurs when a quantity decreases at a rate proportional to its current value, which is the opposite of exponential growth.
For exponential decay, such as \( 2^{-x} \):
For exponential decay, such as \( 2^{-x} \):
- **Rapid Decrease**: As \( x \) increases, \( 2^{-x} \) quickly approaches zero, contributing a damping effect to the growth of the polynomial part.
- **Dominant Effect Over Time**: Eventually, this decay becomes the dominant factor in the behavior of the graph, causing it to decline sharply after an initial rise.
- **Infinity Limitations**: Even if the polynomial part rises initially, it will eventually be outweighed by the decay, leading the function to decrease as \( x \) goes to infinity.
Behavior of Graphs
The behavior of the graph of \( y = x^n \cdot 2^{-x} \) demonstrates a complex interplay between the polynomial and exponential components. Observing the combined effect of both parts is key to understanding how the graph behaves as \( x \) varies.
Here is what typically happens for different \( n \) values:
Here is what typically happens for different \( n \) values:
- **Initial Increase**: With smaller \( x \) values, the polynomial component often causes the function to initially rise. The rate and peak position of this rise depend on the power \( n \).
- **Peaking and Decline**: As \( x \) continues to increase, the exponential decay starts to overpower the polynomial growth, causing the graph to peak and then decline. The graph's peak becomes more pronounced with higher \( n \) values.
- **Graph Shape**: The starting peak of the graph shifts closer to \( x = 0 \) with higher \( n \), while the height of the peak increases, reflecting the stronger initial effect of \( x^n \).
Parameter Analysis
Parameter analysis involves examining how changes in parameters鈥攈ere, the value of \( n \)鈥攁ffect the overall function \( y = x^n \cdot 2^{-x} \) and its graph. By analyzing the parameter \( n \), we deeply understand its role in shaping the function.
Important points for parameter values:
Important points for parameter values:
- **Effect of Different \( n \) Values**: As \( n \) increases, the initial growth of the function becomes faster due to the increased degree of the polynomial. Each increase in \( n \) provides a sharper initial rise in the graph before the effects of exponential decay take over.
- **Peak Characteristics**: Higher \( n \) values mean the function peaks at higher \( y \)-values, and these peaks occur closer to \( x=0 \).
- **Graphical Implications**: Analyzing how \( n \) influences these characteristics helps predict the function's behavior without actual graphing, providing valuable insights for anticipating the graph's structure and changes.
