Question

Create 2 parabolas with a smooth transition at P, out of the figure below.
This is what I did:
The first one I found using this technique.
P1 = k(x+10)(x-10)+10
15 = k(0+10)(0-10)
Removing 10 from y axis so I can treat the (10,10) as a zero (10,0).
This gives P1 = -0.15x^2+25
P1 ' (10) = -3

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P2 was trickier.
I solved it like this:
P2 = ax^2 + bx + c
100a + 10b + c = 10
P2 at 10 should be 10.

20a + b = -3
The derivative at 10 should be -3

b^2 = 4ac
It has a single zero.
I got the right solution but the method seems wrong to me. How would you solve it?

Answer 1

The answer is in the form:
\[\frac{9}{4} (x - \frac{50}{3})^{2}\]

Answer 2

9 / 40 *