91Ó°ÊÓ

The vertex of a parabola is the highest or lowest point on the graph, depending on whether the parabola opens upwards or downwards, respectively. For the given quadratic function \( y=-\frac{4}{9}x^2 + \frac{24}{9}x + 12 \), since the coefficient of the \( x^2 \) term is negative, the parabola opens downwards, and thus the vertex represents the maximum point.
To find the vertex, we can use the formula \( x=-\frac{b}{2a} \) where \( a \) and \( b \) are the coefficients from the quadratic equation in the form \( y=ax^2+bx+c \). This formula gives us the x-coordinate of the vertex after substituting the identified coefficients. Following this method ensures a straightforward process to pinpoint the parabola’s peak or valley.
Once the x-coordinate is calculated, substituting this value back into the original equation gives us the y-coordinate, thus completing the vertex's coordinates \( (x, y) \), which in this exercise specifically yields \( (3, 14) \).

In the context of projectile motion, such as a diver's path, the maximum height is the greatest y-value the object reaches. For quadratic functions like our example, this maximum height corresponds to the y-coordinate of the vertex when the parabola opens downwards.
Once the vertex is located using the aforementioned vertex formula, the y-value at this point is calculated by substituting the x-coordinate (in this case, 3 feet) back into the original function. The resulting y-value gives you the maximum height the diver reaches above the water surface, representing the apex of the dive. For the diver’s path defined by \( y=-\frac{4}{9}x^2 + \frac{24}{9}x + 12 \), the computation shows that the maximum height reached is 14 feet.

Factoring quadratics is a powerful tool for solving quadratic equations. It involves breaking down the quadratic into a product of binomials. However, in the case of our example \( y=-\frac{4}{9}x^2 + \frac{24}{9}x + 12 \), the coefficients are not ideal for simple factorization.
In some cases, if the quadratic is factorable over the integers, it can be expressed in the form \( y=(mx + n)(px + q) \), where \( m, n, p, \) and \( q \) are integers. Factoring is beneficial for finding the roots of the parabola—the x-values where the graph intersects the x-axis—but it is not necessary for finding the vertex or maximum height. Instead, as shown in the exercise, using the vertex formula and substituting the resulting x-coordinate of the vertex into the original equation proves to be an efficient method for these particular calculations.