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The given function is
[tex]f(x)=0.5(x+3)(x-7)[/tex]First, we have to find the vertex. So, let's solve the product.
[tex]\begin{gathered} f(x)=0.5(x^2-7x+3x-21) \\ f(x)=0.5(x^2-4x-21) \\ f(x)=0.5x^2-2x-10.5 \end{gathered}[/tex]Where a = 0.5 and b = -2. Let's find the horizontal coordinate of the vertex
[tex]h=-\frac{b}{2a}=-\frac{-2}{2\cdot0.5}=\frac{2}{1}=2[/tex]Then, we find the vertical coordinate of the vertex
[tex]k=0.5(2+3)(2-7)=0.5\cdot5(-5)=-25\cdot0.5=-12.5[/tex]The important thing about the vertex is that the coordinate k tells us the maximum or minimum. In this case, the function has a minimum at -12.5 because that's the lowest point reached by the function. The image below shows the graph
