Answer to Question #175224 in Linear Algebra for Tush

Question

Determine the polynomial function whose graph passes through the points (0, 1), (2,1/3) and (4,1/5). Also sketch the graph of the polynomial function. 

Answer 1

Solution.

Each of the points $(0,1), (2,\frac{1}{3}),(4,\frac{1}{5})$ satisfies the equation $p=ax^2+bx+c$ for some unknown $a, b, c.$  Substitute each point in the equation and make a matrix equation:

$$AX=B, \text{or}$$ $$\begin{pmatrix} 0 & 0 &1 \\ 4 & 2 & 1\\ 16 & 4 & 1 \end{pmatrix}\cdot \begin{pmatrix} a \\ b\\ c \end{pmatrix} =\begin{pmatrix} 1 \\ \frac{1}{3}\\ \frac{1}{5} \end{pmatrix}$$

Solve matrix equation: $X = A^{-1} · B.$

$X=\begin{pmatrix} \frac{1}{15} \\ -\frac{7}{15}\\ 1 \end{pmatrix},$ or $a=\frac{1}{15}, b=-\frac{7}{15}, c=1.$

So, $p=\frac{1}{15}x^2-\frac{7}{15}x+1.$ It is the polynomial function.