Question

Graph the functions \(8 n, 4 n \log n, 2 n^{2}, n^{3},\) and \(2^{n}\) using a logarithmic scale for the \(x\) - and \(y\) -axes. That is, if the function is \(f(n)\) is \(y,\) plot this as a point with \(x\) -coordinate at \(\log n\) and \(y\) -coordinate at \(\log y\)

Short answer

Convert each function point to \((\log n, \log f(n))\) and plot them on a logarithmic scale graph.

Step-by-step solution

Identify the Functions

The functions to graph are: \(8n\), \(4n \log n\), \(2n^{2}\), \(n^{3}\), and \(2^{n}\).

Determine Coordinates

For each function \(f(n)\), calculate points where \(n\) takes specific values (such as 1, 2, 4, 8, etc.). Then, convert these coordinates into their logarithmic forms, \((\log n, \log f(n))\).

Calculate \(\log n\) and \(\log f(n)\) for Each Point

Here are a few points:For \(8n\):\(n = 1\), \((\log 1, \log 8) = (0, \log 8)\)\(n = 2\), \((\log 2, \log 16) = (\log 2, \log 16)\)For \(4n\log n\):\(n = 2\), \((\log 2, \log 4n \log n) = (\log 2, \log(4 \times 2 \log 2))\)Etc.

Graph the Functions Using Logarithmic Scale

On graph paper or using graphing software, plot the points obtained in the previous step for each function. Use a logarithmic scale for both axes.

Key concepts

logarithmic scale

A logarithmic scale is a type of scale used for a large range of values. It uses the logarithmic function to compress or stretch the data. This scale is useful when you are dealing with values that span several orders of magnitude. Instead of plotting the actual values, you plot their logarithms. This is why you'll often see graphs with axes labeled as \(log n\) and \(log f(n)\).
Logarithmic scales are especially helpful in visualizing data like population growth or comparing different algorithms as shown in the exercise. The logarithmic scale can make exponential growth look linear and large ranges easier to compare.
Suppose we have 8n and 2^n functions. On a normal scale, 2^n would quickly outpace 8n, but on a logarithmic scale, we can more easily compare their behaviors even for large n.

complexity analysis

Complexity analysis is an important part of computer science that helps us understand the efficiency of algorithms. It measures how the runtime or space requirements of an algorithm grow as the input size increases.
The functions in the exercise, such as \(4n \, log \, n\) and \(2n^2\), represent the time complexities of different algorithms. By graphing these complexities, we can visually see the differences in their growth rates.
Linear time complexity (like 8n) grows steadily with input size. Polynomial time complexity (like 2n^2) grows much faster. Exponential time (like 2^n) grows the fastest and becomes impractical for large inputs. Complexity analysis allows us to identify which algorithms are more scalable and suitable for large datasets.

growth of functions

Understanding the growth of functions is crucial for predicting the performance of algorithms. In the exercise, you are asked to compare functions like \(n^3\) and \(8n\).
Different functions grow at different rates:

• Linear growth: 8n grows at a constant rate.
• Log-linear growth: 4n \, log \, n grows slightly faster due to the \ log n\ factor.
• Quadratic growth: 2n^2 grows even faster since changes in n result in squared changes in the function.
• Cubic growth: n^3 grows more quickly, with growth rate increasing cubically as n increases.
• Exponential growth: 2^n is the fastest among these, doubling with each increment in n.

By comparing these in a logarithmic scale, you can observe how each type of function grows. This is invaluable for choosing the right algorithms that balance performance and scalability.

graphing functions

Graphing functions can help you visually understand and compare their growth rates. In the exercise, you plot the functions using a logarithmic scale.
Here’s how to proceed:

• Identify the functions: List all functions you need to graph (e.g., 8n, 4n \ log \ n, 2n^2, n^3, 2^n).
• Calculate coordinates: For each function, pick values for n (like 1, 2, 4, 8), then compute the corresponding \(f(n)\) and take their logarithms.
• Plot points: Use the coordinates \(( \,log n, \, log f(n) \, )\) you calculated and plot these points on a graph with logarithmic scales on both axes.
• Connect the points: Draw the curves connecting the plotted points to clearly see how each function grows.

This method helps reveal how different algorithms perform as input sizes become large, making it easier to grasp their complexities.