Given three points (-1,4) (1,6) (2,10) how do you write a quadratic function in standard form with the points?
Solve a system of three equations in three unknowns.
Let us assume that the axis of symmetry for the parabola is vertical.
The standard equation of a parabola with a vertical axis is
Using the coordinates of the above points, we know that...
This simplifies to
We also have:
The three equations give us the following system, which has a unique solution (a, b, c):
While we may solve these using any method, it will be convenient to subtract equation (1) from equation (2).
Change the signs and add:
Adding gives us:
Our system is now reduced to just two unknowns. Putting the known value of b into the three equations gives:
Subtract the first from the second to obtain the equation:
Substitute a = 1 into the equation
The equation of our parabola has a = 1, b = 1, and c = 4.
Therefore, it is
Check, and you will find that all three points fit the equation.