# Do you know of any software I can use to graph this equation?

• karpmage
In summary, the individual is explaining their thought process behind graphing a curve with a parabolic directrix and a focus point. They express their equation and use Cardano's method to solve it. They then ask for suggestions on software to plot the curve and later find an answer on a different forum.
karpmage
I thought I'd just quickly tell you guys why I want to graph this equation before giving it you.

We're studying conic sections at the moment, and I started wondering what would happen if I let the directrix be a parabola. I looked at the instance where the directrix is y=x^2. I let the focus, F, be (x_F,y_F). Let P=(x,y) be a point on the curve that I am trying to find. I was wondering what the equation of this curve would be if I said that the distance between P and F and the perpendicular distance between P and the directrix has to be the same for any point P. ie. if we let O be the point where P meets the directrix at right angles, then OP=PF. I shall let O=(x_1,y_1)

The gradient of the directrix at point O has to 2(x_1). Thus the gradient of OP has to be (-1/(2(x_1))). Thus, we can represent O as (x+t, y-(t/(2(x_1))), where t is an unknown variable. Since x_1=x+t, we can further represent O as (x+t, y-(t/(2(x+t))). Substituting these values for x_1 and y_1 back into our equation for the directrix, we get:

y-(t/(2(x+t))=(x+t)^2

I let u=x+t, now: y-(u-x)/(2u)=u^2 => 2uy-(u-x)=u^3 => u^3+(1-2y)u-x=0

I then solve this cubic using Cardano's method (method in the link below) http://www.math.cornell.edu/~henderson/courses/M403-S03/cubics.htm

I get u=(((x/2)+((x/2)^2+((1-2y)/3)^3)^0.5)^(1/3))+(((x/2)-((x/2)^2+((1-2y)/3)^3)^0.5)^(1/3)) thus t=(((x/2)+((x/2)^2+((1-2y)/3)^3)^0.5)^(1/3))+(((x/2)-((x/2)^2+((1-2y)/3)^3)^0.5)^(1/3))-x

I want to show that OP=PF. (PF)^2= (x_F-x)^2 + (y_F-y)^2, and (OP)^2 = (y-y_1)^2 + (x-x_1)^2 = (t/(2(x+t))^2 + t^2

Thus, replacing t in terms of x and y we get: (((((x/2)+((x/2)^2+((1-2*y)/3)^3)^0.5)^(1/3))+(((x/2)-((x/2)^2+((1-2*y)/3)^3)^0.5)^(1/3))-x)^2)*((((x/2)+((x/2)^2+((1-2*y)/3)^3)^0.5)^(1/3))+(((x/2)-((x/2)^2+((1-2*y)/3)^3)^0.5)^(1/3))+1)/(4*((((x/2)+((x/2) ^2+(1-2*y)^3)^0.5)^(1/3))+(((x/2)-((x/2)^2+(1-2*y)^3)^0.5)^(1/3)))^2)-(x_F-x)^2-(y_F-y)^2 = 0

(easier to see equation)

This is the function that I want to plot. I'm just interested to see what it'd look like. Of course I'd have to replace x_F and y_F with set values. I just want to know if any of you know of software which can plot this equation. I don't want to plot a 3-D graph though, unless I am able to set the value of z to 0.

OR

If you guys know of a way to rearrange this equation so that y becomes the subject, then I can simply use any graphing software to plot the curve.

## 1. What is the best software to use for graphing equations?

There are many software options available for graphing equations, but some popular choices include Microsoft Excel, Wolfram Alpha, and Desmos. It ultimately depends on your specific needs and preferences.

## 2. Can I graph multiple equations at once using this software?

It depends on the software you are using. Some programs like Microsoft Excel allow you to graph multiple equations on the same graph, while others may require you to create separate graphs for each equation.

## 3. Is this software easy to use for beginners?

Many graphing software programs offer user-friendly interfaces and tutorials to help beginners get started. Some may be more complex than others, so it's important to read reviews and try out different options to find the one that works best for you.

## 4. Can I input my own custom equations into this software?

Most graphing software programs allow you to input custom equations, but it's always a good idea to check the program's capabilities before purchasing or using it.

## 5. Is there a cost associated with using this software?

Some graphing software programs may require a one-time purchase or a subscription fee, while others may be free to use. It's important to research the cost of the software before deciding which one to use.

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