Question

Can someone help me please

Answer 1

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 $x$-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 $x$-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 $(x-3)(x-7)=0$, which gives $x-3=0 \vee x-7=0$ (the 'V' stands for 'or'). We can take the average of the two roots to get the $x$-coordinate of the minimum point of the graph (which, in this case, is $x=\frac{3+7}{2} = \frac{10}{2} = 5$).

Having the $x$-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 $y$-coordinate of the extremum.