Question

Write an equation (any form) for the quadratic graphed below:

Answer 1

Check the picture below.

so the function touches the x-axis at -1, hmm however, let's notice something, it doesn't cross it, it simply hits the x-axis and bounces off of it, well, that means it has a root at -1 with an even multiplicity, hmm let's settle for 2, so we can say it has a root at -1 with multiplicity of 2. We can also see it passes through (-3 , 2).

Now, we can say, what's the equation of a function with root of -1 of multiplicity of 2, that also passes through (-3 , 2)?

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