Answer:
"I can find the maximum or minimum by looking at the factored expression of a quadratic function by reading off its roots and taking the arithmetic average of them to obtain the [tex]x[/tex]-coordinate of the quadratic function, and then substituting that value into the function."
Step-by-step explanation:
Because of the symmetry of quadratics (which is the case here because we have two factors of degree 1, so we are dealing with a polynomial of degree 2, which is a fancy way of saying that something is a quadratic), the [tex]x[/tex]-coordinate of the extremum (a fancy way of saying maximum or minimum) is the (arithmetic) average of the two roots.
In the factored form of a quadratic function, we can immediately read the roots: 3 and 7. Another way to see that is to solve [tex](x-3)(x-7)=0[/tex], which gives [tex]x-3=0 \vee x-7=0[/tex] (the 'V' stands for 'or'). We can take the average of the two roots to get the [tex]x[/tex]-coordinate of the minimum point of the graph (which, in this case, is [tex]x=\frac{3+7}{2} = \frac{10}{2} = 5[/tex]).
Having the [tex]x[/tex]-coordinate of the extremum, we can substitute this value into the function to obtain the maximum or minimum point of the graph, because that will give the [tex]y[/tex]-coordinate of the extremum.
