Just for fun: Cubic Graph Plot

In summary, the conversation discusses the use of Desmos.com, an online graphing utility, to experiment with the Newton-Raphson method for solving cubic graphs. The graph shows the turning points and intercepts of the cubic graph, with the x-intercepts being calculated using the method. It is suggested that readers may need to add more iterations to improve the accuracy of the x-intercepts for certain parameter values. The conversation also introduces another use of Newton-Raphson for finding the square root of a number, with the accurate numeric value of the root shown on the side panel. The starting value for this method is the square root of the nearest square greater than the given number.
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neilparker62
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Cubic Graph plot on Desmos
Desmos.com is a great online graphing utility which I'm sure is familiar to many PF users. I wanted to experiment with the Newton-Raphson method using it so chose solution of cubic graphs as an example. The graph shows a variable cubic on which all turning points and intercepts are calculated and shown. x-intercepts by Newton-Raphson.

I'm sure readers will find parameter values where the x-intercepts are not that accurate - in that case add a couple more "g(g(.." iterations!

https://www.desmos.com/calculator/lejtdd8fws
 
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Here's another use of Newton-Raphson whereby the square root of a number is determined by finding the positive x-intercept of ##f(x)=x^2-a##. The intercept is marked on the graph but the side panel shows a more accurate numeric value of the root against parameter d where ##d=\sqrt{a}##. The start value for determination of the square root is the square root of the nearest square greater than a.

https://www.desmos.com/calculator/qbyev8tuqz
 
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1. What is a cubic graph plot?

A cubic graph plot is a type of graph that shows the relationship between two variables, where one variable is raised to the power of three (cubed). It is represented by a curved line, and is commonly used to analyze mathematical equations and functions.

2. How do you create a cubic graph plot?

To create a cubic graph plot, you will need to have a set of data points for both variables. Then, using a graphing calculator or software, you can plot the points and connect them with a smooth curve. Alternatively, you can also use a ruler and graph paper to manually plot the points.

3. What are the key features of a cubic graph plot?

The key features of a cubic graph plot include the shape of the curve, the direction of the curve, and the x and y intercepts. The shape of the curve can be either concave up or concave down, depending on the equation. The direction of the curve can indicate whether the function is increasing or decreasing. The x and y intercepts are points where the curve crosses the x and y axes, respectively.

4. How is a cubic graph plot useful in science?

A cubic graph plot is useful in science for analyzing and understanding mathematical relationships between variables. It can also be used to model and predict real-world phenomena, such as growth rates, chemical reactions, and physical processes. Additionally, cubic graph plots can help identify patterns and trends in data, which can aid in making scientific conclusions and predictions.

5. What are some common misconceptions about cubic graph plots?

One common misconception about cubic graph plots is that the curve must always pass through the origin (0,0). While this may be true for some cubic functions, it is not a general rule. Another misconception is that the slope of the curve is always constant, when in fact it can vary at different points along the curve. It is important to carefully analyze the data and equation to fully understand the behavior of a cubic graph plot.

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