Question

Given three points (-1,4) (1,6) (2,10) how do you write a quadratic function in standard form with the points?

Answer 1

Solve a system of three equations in three unknowns.

Explanation:

Let us assume that the axis of symmetry for the parabola is vertical.

The standard equation of a parabola with a vertical axis is `y = ax^2 + bx + c`, where a, b, and c are real numbers -- with `a ne 0`.

Using the coordinates of the above points, we know that...
`a(-1)^2 + b(-1) + c = 4`
This simplifies to `a - b + c = 4`.

We also have:
`a1^2 + b(1) + c = 6`
That is, `a + b + c = 6`.

And finally:
`a(2)^2 + b(2) + c = 10`
`4a + 2b + c = 10`

The three equations give us the following system, which has a unique solution (a, b, c):

`a - b + c = 4`
`a + b + c = 6`
`4a + 2b + c = 10`.

While we may solve these using any method, it will be convenient to subtract equation (1) from equation (2).

`a + b + c = 6`
`a - b + c = 4` (subtract)

Change the signs and add:
`a + b + c = 6`
`-a + b - c = -4` (add)

Adding gives us:
`2b = 2`
`b = 1`.

Our system is now reduced to just two unknowns. Putting the known value of b into the three equations gives:

`a - 1 + c = 4`
`a + 1 + c = 6`
`4a + 2 + c = 10`.

Simplify to:
`a + c = 5`
`4a + c = 8`.

Subtract the first from the second to obtain the equation:
`3a = 3`
`a = 1`

Substitute a = 1 into the equation `a + c = 5` and obtain `c = 4`.

The equation of our parabola has a = 1, b = 1, and c = 4.

Therefore, it is
`y = x^2 + x + 4`
Check, and you will find that all three points fit the equation.