Question

PLEASE HELP
Write a function in any form that would match the graph shown below

Answer 1

looking at the picture above, hmmm we can see the graph touches at -5 and bounces right back up, that means even multiplicity, hmmm we can use say multiplicity of 2 for that, so hmm the graph also touches at 4 and bounces right back, same multiplicity.

so we can say, what's the equation of a function with roots at x = -5 and x = 4 and with multiplicity of 2 on each, that also passes through the point (0 , 80) ?

`\begin{cases} x = -5 &\implies x +5=0\\ x = -5 &\implies x +5=0\\ x = 4 &\implies x -4=0\\ x = 4 &\implies x -4=0 \end{cases}\hspace{5em}\boxed{\textit{each with multiplicity of 2}} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +5 )( x +5 )( x -4 )( x -4 ) = \stackrel{0}{y}} \hspace{5em}\textit{we also know that } \begin{cases} x=0\\ y=80 \end{cases}`

`a ( 0 +5 )( 0 +5 )( 0 -4 )( 0 -4 ) = 80\implies \implies a(25)(16)=80\implies 400a=80 \\\\\\ a=\cfrac{80}{400}\implies a=\cfrac{1}{5}\hspace{9em} {\Large \begin{array}{llll} \cfrac{1}{5}(x+5)^2(x-4)^2 = y \end{array}}`

Check the picture below.