Question

Use the graph to write the formula for a polynomial function of least degree.

Answer 1

we do this one, the same way as the one previously, so hmmm Check the picture below, it has zeros or roots at -2 , 1 and 3, meaning is a cubic equation, and it passes through (0 , 3) up above as you see in the picture.

`\begin{cases} x = -2 &\implies x +2=0\\ x = 1 &\implies x -1=0\\ x = 3 &\implies x -3=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +2 )( x -1 )( x -3 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that} \begin{cases} x=0\\ y=3 \end{cases} \\\\\\ a ( 0 +2 )( 0 -1 )( 0 -3 ) = 3\implies 6a=3\implies a=\cfrac{3}{6}\implies a=\cfrac{1}{2} \\\\[-0.35em] ~\dotfill`

`\cfrac{1}{2}( x +2 )( x -1 )( x -3 )=y\implies \cfrac{1}{2}( x +2 )(x^2-4x+3)=y \\\\\\ \cfrac{1}{2}(x^3-2x^2-5x+6)=y\implies \boxed{\cfrac{x^3}{2}-x^2-\cfrac{5x}{2}+3=y}`