Question

Write the equation of a quadratic function that contains the points (1,21), (2,18), and (-1,9)

Answer 1

The equation of a quadratic function that contains the points (1, 21), (2,18) and (-1, 9) is
`y = -3\cdot x^(2)+6\cdot x +18`.

Explanation:

A quadratic function is a second order polynomial of the form:

`y = a\cdot x^(2)+b\cdot x + c` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`a`,
`b`,
`c` - Coefficients.

From Algebra we understand that a second order polynomial is determined by knowing three distinct points. If we know that
`(x_(1), y_(1)) = (1, 21)`,
`(x_(2),y_(2)) = (2,18)` and
`(x_(3), y_(3)) = (-1, 9)`, then we construct the following system of linear equations:

`a+b+c = 21` (2)

`4\cdot a + 2\cdot b + c = 18` (3)

`a - b + c = 9` (4)

By algebraic means, the solution of the system is:

`a = -3`,
`b = 6`,
`c = 18`

Therefore, the equation of a quadratic function that contains the points (1, 21), (2,18) and (-1, 9) is
`y = -3\cdot x^(2)+6\cdot x +18`.