The set of points given in coordinate form must be a function for the ideas covered in the following methods. This means that no two points in the set have the same first coordinate and the given number of points are distinct. All sets of points used in this Web page will be a function.

In general:

two distinct points will determine a straight line or linear function,

three distinct points will determine a quadratic function

(assume the three points do not lie on a straight line),

four distinct points will determine a cubic function

(assume the four points do not lie on a straight line or on a quadratic),

and so on.

The methods used in the following examples assumes a knowledge of solving systems of equations using matrix solutions. Review this method in your algebra course if you need more information.

In general, a matrix equation consisting of a constant matrix ** A** times a variable matrix

In practice, ** A^(-1)**, the inverse of constant matrix

EXAMPLES:

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If points (-1, 1) and (0, 3) are given as points on a linear function:

then we can find ** a** and

Replace x and y with the coordinate values given to get the system:

Note that *b* = *b*** (** 1

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (-1, 1) and (0, 3) are given as points on a linear function then:

is the linear function.

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If points (1, 1), (2, 5) and (-1, -1) are given as points on a quadratic function:

then we can find ** a**,

Note that FUNction 2. can be written:

Replace x and y with the coordinate values given to get the system:

Note that *c* = *c*** (** 1

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (1, 1), (2, 5) and (-1, -1) are given as points on a quadratic function then:

is the quadratic function.

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If points (1, -3), (-1, -1), (2, -13) and (-2, -21) are given as points on a quadratic function:

then we can find ** a**,

Note that FUNction 3. can be written:

Replace x and y with the coordinate values given to get the system:

Note that *d* = *d*** (** 1

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (1, -3), (-1, -1), (2, -13) and (-2, -21) are given as points on a cubic function then:

is the cubic function.

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If points (-1, 10), (1, 4), (2, 10), (3, 22) and (4, 10) are given as points on a quartic function:

then we can find ** a**,

Replace x and y with the coordinate values given to get the system:

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (-1, 10), (1, 4), (2, 10), (3, 22), (4, 10) and (9, 3372) are given as points on a quartic function then:

is the quartic function.

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If points (0, -288), (1, 12), (3, 0), (7, 192), (8, 1020) and (9, 3372) are given as points on a quintic function:

then we can find ** a**,

Replace x and y with the coordinate values given to get the system:

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (0, -288), (1, 12), (3, 0), (7, 192), (8, 1020) and (9, 3372) are given as points on a quintic function then:

is the quintic function.

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A comment about notation for this n - points section.
** an** is one symbol with the subscript

If this is your first or second time using subscript notation then it is best if you copy the following section in your handwriting and move the superscrips, ^, up as exponents and the subscripts down in the normal subscript line.

If you do this writing you may find it easer to see and understand if your superscripts and subscripts are a bit smaller font size than the variable they are associated with.

If points (x1, y1), (x2, y2), (x3, y3) . . . and (xn, yn) are given as points on an (n-1) degree polynomial function:

then we can find ** an**,

Replace x and y with the coordinate values given to get the system:

Note that *a1* = *a1*** (** 1

The matrix equation for this system is:

Then finding the inverse of the coefficient matrix gives:

Therefore the solution of the matrix equation is found by:

The solution of the matrix equation is:

If points (x1, y1), (x2, y2), (x3, y3) . . . and (xn, yn) are

given as points on an (n-1) degree polynomial function:

is the (n-1) degree polynomial function.

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