The problem asks us to analyze the quadratic equation $y = 6 + 4x - x^2$. We need to find the vertex of the graph of the equation, determine if the vertex is a maximum or minimum point, determine the value of $x$ that gives the optimal value of the function, and determine the optimal (maximum or minimum) value of the function.

1. Problem Description

The problem asks us to analyze the quadratic equation $y = 6 + 4x - x^2$. We need to find the vertex of the graph of the equation, determine if the vertex is a maximum or minimum point, determine the value of $x$ that gives the optimal value of the function, and determine the optimal (maximum or minimum) value of the function.

2. Solution Steps

(a) Find the vertex of the graph of the equation $y = 6 + 4x - x^2$.

First, rewrite the equation in the standard form of a quadratic equation: $y = ax^2 + bx + c$.

$y = -x^2 + 4x + 6$

In this case, $a = -1$, $b = 4$, and $c = 6$.

The x-coordinate of the vertex is given by the formula:

$x = -\frac{b}{2a}$

$x = -\frac{4}{2(-1)} = -\frac{4}{-2} = 2$

Now, substitute the value of $x$ into the equation to find the y-coordinate of the vertex:

$y = -(2)^2 + 4(2) + 6$

$y = -4 + 8 + 6$

$y = 10$

So, the vertex of the graph is $(2, 10)$.

(b) Determine whether the vertex is a maximum or minimum point.

Since the coefficient of the $x^2$ term ($a$) is negative ($a = -1$), the parabola opens downwards. This means that the vertex represents a maximum point.

(c) Determine what value of $x$ gives the optimal value of the function.

The optimal value occurs at the vertex. The x-coordinate of the vertex is $x=2$.

(d) Determine the optimal (maximum or minimum) value of the function.

The optimal value is the y-coordinate of the vertex, which is $y=10$.

3. Final Answer

(a) The vertex of the graph is $(2, 10)$.

(b) The vertex is a maximum point.

(c) The value of x that gives the optimal value of the function is $x = 2$.

(d) The optimal (maximum) value of the function is $y = 10$.