Solving Cubic Equations: Finding a Line Through 2 Points

  • Thread starter programmer
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In summary: Mm... that's how all x's are typeset in every math book I've ever seen.In summary, you can find a cubic equation that passes through two points by solving for a.
  • #1
programmer
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I've taken Calculus 1, but it was a few years ago, so bear with me. I understand how to use derivitaves to find critical numbers, relative max's and min's, points of inflections, incresing, decreasing. all that good stuff.

my question is, if I have two points I want a x^3 line pass through, how can i accomplish this?
 
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  • #2
programmer said:
my question is, if I have two points I want a x^3 line pass through, how can i accomplish this?
y = x3 is a curve, not a line.
I don't really get what you mean. y = x3 is a specific curve, i.e you cannot change the set of points that it passes through.
If you want to get 2 points on the curve, just choose 2 arbitrary x1, and x2 values, then from there find the corresponding y1, and y2. And you'll have 2 points that the curve passes through.
 
  • #3
no, i want to MODIFY a x^3 equation to MAKE it pass through the points i already have set.
 
  • #4
the first point is 0,0, and the 2nd point could be anything, how to i alter the curve to make it pass through my 2nd point?
 
  • #5
You're trying to curve-fit a cubic to some known data set.

Look up cubic spline interpolation. Numerical Recipes in C has a section on it, with the algorithm coded in C.

http://www.library.cornell.edu/nr/cbookcpdf.html [Broken]

If you only have two points, one of which is (0, 0) and the other is (x0, y0), just solve this equation for a:

[itex]a x_0^3 = y_0[/itex]

- Warren
 
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  • #6
There are many different ways to do this. For example, as chroot said, if you want a formula of the form y= ax3, which necessarily passes through (0,0) for all a, just select a so that y0= ax3: That is [tex]y= \left(\frac{y_0}{x_0^3}\right)x^3[/tex] passes through (0,0) and (x0,y0).

Or, you could alter y= x3 to look like y= x3+ ax. In order to have y= y0 when x= x0 we must have [itex]y_0= x_0^3+ ax_0[/itex] or, solving for a, [itex]a= \frac{y_0- x_0^3}{x_0}[/itex]. That is, the graph of [itex]y= x^3+ \frac{y_0- x_0^3}{x_0}x[/itex] passes through (0,0) and (x0,y0). There are many other possiblities. The choice is essentially arbitrary unless you have other conditions to fulfill.
 
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  • #7
well...i guess I've forgotten a lot of calculus...what does the I in those equations stand for? Integral?
 
  • #8
programmer said:
well...i guess I've forgotten a lot of calculus...what does the I in those equations stand for? Integral?
?? What I are you talking about? Which response does this relate to?
 
  • #9
nevermind...the X's in those equations look like I's...

that's retarded
 
  • #10
programmer said:
nevermind...the X's in those equations look like I's...

that's retarded

Mm... that's how all x's are typeset in every math book I've ever seen.

- Warren
 

1. How do you find the equation of a line through two given points?

To find the equation of a line through two points, you can use the slope-intercept form y = mx + b, where m is the slope and b is the y-intercept. First, calculate the slope using the formula (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the coordinates of the two points. Then, substitute the slope and one of the points into the equation to solve for b. This will give you the equation of the line in the form y = mx + b.

2. What is the significance of finding a line through two points in solving cubic equations?

Finding a line through two points is crucial in solving cubic equations because it helps us determine the relationship between the x and y coordinates of the points. This relationship is represented by the equation of the line and can be used to find the solutions to the cubic equation.

3. Can you use the slope-intercept form to find the equation of a line through two points in a cubic equation?

Yes, you can use the slope-intercept form to find the equation of a line through two points in a cubic equation. This form is a standard method for finding the equation of a line and can be used for any type of equation, including cubic equations.

4. Are there any other methods for finding the equation of a line through two points?

Yes, there are other methods for finding the equation of a line through two points. These include the point-slope form, where you use the slope and one point to find the equation, and the two-point form, where you directly plug in the coordinates of the two points into the equation. However, the slope-intercept form is the most commonly used method.

5. How does finding a line through two points help in solving cubic equations graphically?

Finding a line through two points helps in solving cubic equations graphically by providing a visual representation of the relationship between the x and y coordinates of the points. This can help in identifying the solutions to the cubic equation, as they will be the points where the line intersects the x-axis. Additionally, the slope of the line can also provide information about the nature of the solutions (real or complex) to the cubic equation.

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