Question

Write an equation of the parabola that has its x intercepts at (-3,0) and (4,0) and its y intercept at (0,-6)

Answer 1

the x-intercepts in a function are pretty much just its roots or solutions, so let's reword that

what's the equation of a parabola with roots at -3 and 4 that it passes through (0 , -6)?

`\begin{cases} x = -3 &\implies x +3=0\\ x = 4 &\implies x -4=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +3 )( x -4 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that } \begin{cases} x=0\\ y=-6 \end{cases} \\\\\\ a ( 0 +3 )( 0 -4 ) = -6\implies -12a=-6\implies a=\cfrac{-6}{-12}\implies a=\cfrac{1}{2} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{1}{2}( x +3 )( x -4 ) =y\implies \cfrac{1}{2}(x^2-x-12)=y\implies \boxed{\cfrac{x^2}{2}-\cfrac{x}{2}-6=y}`

Check the picture below.