Question #175224

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. 


1
Expert's answer
2021-03-26T08:39:07-0400

Solution.

Each of the points (0,1),(2,13),(4,15)(0,1), (2,\frac{1}{3}),(4,\frac{1}{5}) satisfies the equation p=ax2+bx+cp=ax^2+bx+c for some unknown a,b,c.a, b, c.  Substitute each point in the equation and make a matrix equation:


AX=B,orAX=B, \text{or}(0014211641)(abc)=(11315)\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=A1B.X = A^{-1} · B.

X=(1157151),X=\begin{pmatrix} \frac{1}{15} \\ -\frac{7}{15}\\ 1 \end{pmatrix}, or a=115,b=715,c=1.a=\frac{1}{15}, b=-\frac{7}{15}, c=1.

So, p=115x2715x+1.p=\frac{1}{15}x^2-\frac{7}{15}x+1. It is the polynomial function.


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