# Parabola Equations

*A projectile is launched at x=0 .y=h with speed V and angle θ. What is the equation of the parabola (no air drag)?

y = ax2 + bx + c

a=(g/2)/(Vcosθ)2 … (g is negative)

b=tanθ

c=h

Given 2 points (x[sub]1[/sub],y[sub]1[/sub]) and (x[sub]2[/sub],y[sub]2[/sub]), find the equation of the parabola that passes through the origin and those 2 points

y = ax2 + bx + c

a=(x[sub]2[/sub]y[sub]1[/sub]-x[sub]1[/sub]y[sub]2[/sub])/(x[sub]1[/sub]2x[sub]2[/sub]-x[sub]1[/sub]x[sub]2[/sub]2)

b=-(x[sub]2[/sub]2y[sub]1[/sub]-x[sub]1[/sub]2y[sub]2[/sub])/(x[sub]1[/sub]2x[sub]2[/sub]-x[sub]1[/sub]x[sub]2[/sub]2)

c=0

Given 2 points (x[sub]1[/sub],y[sub]1[/sub]) and (x[sub]2[/sub],y[sub]2[/sub]), find the equation of the parabola that passes through (0,h) and those 2 points

y = ax2 + bx + c

a=(x[sub]1/sub+x[sub]2[/sub]y[sub]1[/sub]-hx[sub]2[/sub])/(x[sub]1[/sub]2x[sub]2[/sub]-x[sub]1[/sub]x[sub]2[/sub]2)

b=-(x[sub]1[/sub]2(h-y[sub]2[/sub])+x[sub]2[/sub]2y[sub]1[/sub]-hx[sub]2[/sub]2)/(x[sub]1[/sub]2x[sub]2[/sub]-x[sub]1[/sub]x[sub]2[/sub]2)

c=h

Given 3 points (x[sub]1[/sub],y[sub]1[/sub]), (x[sub]2[/sub],y[sub]2[/sub]), and (x[sub]3[/sub],y[sub]3[/sub]) find the equation of the parabola that passes through those 3 points

It gets messy.

Given parabola y = ax2 + bx + c , where a<0, find the value of x and y at the apex

x = -b/(2a) … y=c-b2/(4a)

Given parabola y = ax2 + bx + c , where a<0, find the value of x and y for which the slope is -1

x = -(1+b)/(2a) … y = c+(b+1)2/(4a)-b(b+1)/(2a)

parabola 3 points.xls (14.5 KB)

parabola 3 points.xls (14.5 KB)

I’m not sure if that’s a challenge, or a dare.

Reps if you find an error.
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*Given the coordinates (x[sub]p[/sub],y[sub]p[/sub]) of the apex of a parabola whose width is W at a distance D below the apex, find the equation of the parabola.
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*Given 2 points (x1,y1) and (x2,y2), and the slope m1 at (x1,y1), find the equation of the parabola.
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have to ask: did you use LaTeX for those pretty pictures?

I use Maxima to work out the math. Then I capture a PNG screenshot of the area of interest.

For post#1, all the equations were created using the editing available in vBulletin.

*Given the coordinates (0,h) of the launch point and (xp,yp) of the apex, find the equation of the parabola.
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*Given the values of a, b, and c in the equation y=ax2+bx+c of a parabola, find the launch speed V and launch angle theta if the launch point is at (0,h).
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I give Ether my highest complements. It takes a certain kind of problem for me to voluntarily work on it for about 4 hours. I worked through the 1st, 5fth, and 6th sub-problems relatively quickly, but the middle three are what are stalling me. I have attempted to solve sub-problems 3 and 4 using both systems of equations and using matricies with Cramer’s rule. My system of equations was having sign issues (more than can be expected considering I am comparing my solution to Ether’s) and my matricies were not working from my rustiness in this area. I will continue to work tonight and post if anything changes.

Once again, props to you, Ether.

I just finished up the solutions problems in the original post, using three pages of paper in the process. Ether, your solutions check out fine to me (albeit I have very well could have missed something). I also see what you mean by the solution being messy. I think the derivation was messier. To anyone wondering, I solved it each time with a set of matrices and Cramer’s Rule. Cramer’s rule is both really cool and useful. I advise you all to go check it out. I may try my hand at proving the others tomorrow. Ether, as I stated previously, this is a very good challenge. Thank you.

Just finished up all the problems and everything checks out!

I solved most of them using row reduction then the rest were solved using MATLAB. For anyone that’s currently taking a class or looking for a refresher in matrices or 2-d kinematics, try these problems out!