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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