Question

What is the max/min of the quadratic equation in factored form: f(x) = 0.5(x+3)(x-7)? Is it a maximum or a minimum? Show your work or explain your reasoning.

Answer 1

The given function is

`f(x)=0.5(x+3)(x-7)`

First, we have to find the vertex. So, let's solve the product.

`\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}`

Where a = 0.5 and b = -2. Let's find the horizontal coordinate of the vertex

`h=-(b)/(2a)=-(-2)/(2\cdot0.5)=(2)/(1)=2`

Then, we find the vertical coordinate of the vertex

`k=0.5(2+3)(2-7)=0.5\cdot5(-5)=-25\cdot0.5=-12.5`

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