If you get a quadratic function f(x) = ax^2 + bx + c with vertex P = (-4, 1), how can you calculate the values of a, b and c?
Answer
Dear Jade,
Since you get the vertex as a given, you know that the derivative of the function at the point x=-4 is equal to 0. The derivative becomes:
2ax+b so that in x=-4 it becomes equal to 0 if b=8a.
Consequently, the function we are looking for now becomes f(x)=ax^2+bx+c=ax^2+8ax+c
Now we express that the vertex in x=-4 is equal to the function value 1:
1=f(-4)=16a-32a+c=-16a+c
Consequently, we find that c=1+16a.
The parabolas that meet your boundary conditions are: f(x)=ax^2+8ax+16a+1. So under your conditions you have an infinite number of parabolas that meet this requirement. But since you indicate that it must represent a top parabola, the choice for a must be a negative number.
It is normal to get an infinite number of solutions. After all, you know that a line goes through 2 points and by extension it goes through 3 different points exactly 1 parabola. You only give 1 point, but you do give a special point, namely an extremum that counts for 2 points. So you need an additional point to uniquely determine the parabola.
Regards,
Kurt
Answered by
Prof. dr. dr. Kurt Barbe
Mathematics, Statistics, Probability, Scientific Arithmetic
Avenue des Pélain 2 1050 Ixelles
http://www.vub.ac.be/
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