Equations of the three parabolas

Homework Statement

Computing the equations of the three parabolas involved

y=A1x^2 + B1

y= A2x^2+B2

y=A3x^2+B3

The Attempt at a Solution

Tried to get the constants A and B but unsuccessful!!

See the attached file for the three parabolas.The dotted points indiacate the change in curvature.

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berkeman
Mentor
Pretty hard to solve for the numbers when there are no number labels on the graphs..... Am I missing something?

Integral
Staff Emeritus
Gold Member
You will need to specify the spline points, that is, where your parabolas meet.

You should also match the slopes at those points.

Next trouble is that you have:

$$A_n x^2 + B_n$$ where n = 1,2,3
But you need:
$$A_n (x - {k_n})^2 + B_n$$ where n = 1,2,3
Where the $k_n$ specify the x coordinate of the apex of the parabola.

Matching the slopes seems to requires:
$$A_n (x_n- k_n) = A_{n+1}(x_n-k_{n+1})$$ for n = 1 2 where $x_n$ is your 2 spline points.

By matching the slopes you will avoid having a corner at the matching point.

Last edited:
The total length is 7 metres.
The first parabola meets the second at 0.7 metres.
The second parabola meets the third at 6.3 metres.

As you see,at the points where the parabolas meet, there is a change in curvature

berkeman
Mentor
Not enough data. What are the x,y points of join?

Integral
Staff Emeritus