Question

Write a quadratic function und standard form whose graph passes through (-8,0) (-2,0) and (-6,4)

Answer 1

since the point (-8,0) and (-2,0) are x-intercepts or zeros or roots of it, let's reword that to

what's the equation of a quadratic with roots at -8 and -2, that also passes through the point (-6 , 4)?

`\begin{cases} x = -8 &\implies x +8=0\\ x = -2 &\implies x +2=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +8 )( x +2 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that } \begin{cases} x=-6\\ y=4 \end{cases}`

`a ( -6 +8 )( -6 +2 ) = 4\implies a(2)(-4)=4\implies -8a=4\implies a=\cfrac{4}{-8} \\\\\\ a=-\cfrac{1}{2}\hspace{9em}-\cfrac{1}{2}( x +8 )( x +2 ) =y \\\\\\ -\cfrac{1}{2}(x^2+10x+16)=y\implies {\Large \begin{array}{llll} -\cfrac{x^2}{2}-5x-8=y \end{array}}`

Check the picture below.