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Magazine Chapter 13: Problem 16
Plot the graphs of the given functions on log-log paper. $$x^{2} y^{3}=16$$
Short Answer
Expert verified
Rewrite the equation as \( y = \left(\frac{16}{x^2}\right)^{1/3} \). On log-log paper, it is a straight line with slope \(-\frac{2}{3}\).
Step by step solution
01
Rearrange the Equation
We start by rearranging the given equation for the graph that we want to plot. The equation given is \(x^2 y^3 = 16\). We want to solve for \(y\) in terms of \(x\). First, rearrange the equation to: \(y^3 = \frac{16}{x^2}\).
02
Isolate \(y\)
Taking the cube root of both sides allows us to solve for \(y\): \[y = \left(\frac{16}{x^2}\right)^{1/3}\].
03
Express in Log-Log Form
Transform the equation to a form that is suitable for log-log plotting. Take the logarithm of both sides to convert it to log-log form: \(\log(y) = \frac{1}{3} \log(16) - \frac{2}{3} \log(x)\).
04
Plot on Log-Log Paper
On the log-log paper, the equation \(\log(y) = -\frac{2}{3} \log(x) + \frac{1}{3} \log(16)\) is a straight line with a slope of \(-\frac{2}{3}\). The y-intercept is \(\frac{1}{3} \log(16)\). Plot several points by choosing values for \(x\) and calculating the corresponding \(y\).
05
Verify and Finalize the Plot
Count the slope \(-\frac{2}{3}\) between points and match it with calculations. Adjust if necessary for accuracy. Verify a few points on both original and transformed equations to ensure consistency.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Graphs
Logarithmic graphs simplify the visualization of relationships between variables that vary across a wide range of values. When plotted on logarithmic scales, exponential relationships transform into straight lines.
This offers two significant advantages: {
This offers two significant advantages: {
- Enhanced visibility for both very large and very small values, making trends in data more apparent.
- Simplicity in identifying and interpreting power laws or exponential growth/decay, common in scientific and financial domains.
Equation Rearrangement
Rearranging an equation is a fundamental step to simplify and prepare it for further transformations, especially when plotting. Consider the function \(x^2 y^3 = 16\):
- The goal is to express it such that one variable is a function of others, facilitating analysis and graphing easily.
- In this context, rearranging means expressing \(y\) solely in terms of \(x\) as \(y^3 = \frac{16}{x^2}\).
- By isolating \(y\), we achieve a form that is more straightforward for both computational and plotting purposes.
Logarithmic Transformation
Logarithmic transformation provides a powerful tool for converting non-linear relationships into linear ones. This transformation involves taking the logarithm of both sides of an equation. For example, transforming \(y^3 = \frac{16}{x^2}\) results in:
\[\log(y) = \frac{1}{3} \log(16) - \frac{2}{3} \log(x) \]The benefits include:
\[\log(y) = \frac{1}{3} \log(16) - \frac{2}{3} \log(x) \]The benefits include:
- Converting complex polynomial functions into linear equations, making them easier to work with textually and visually.
- Facilitating the application of linear regression techniques on transformed data for better trend analysis.
- Enhancing dynamic range compression, which helps in analyzing multiplicative processes or rates of change.
Plotting on Log-Log Paper
Log-log paper allows for the direct visualization of power-law relationships between variables on a graph. The log-log plot transforms the equation \(\log(y) = -\frac{2}{3} \log(x) + \frac{1}{3} \log(16)\) into a straight line. The key points to remember while plotting include:
- Understanding that the slope of the line, \(-\frac{2}{3}\), indicates the rate at which \(y\) changes with \(x\).
- Recognizing that the y-intercept \(\frac{1}{3} \log(16)\) represents the logarithm of the value for \(y\) when \(x\) equals 1.
- Utilizing log-log plots to easily interpret the proportional relationship and confirm alignment with expected theoretical models.
