The path of a diver is given by the function f(x)=-(4)/(9) x^2+(24)/(9) x+12 where f(x) is the h

step 1 Okay, so this is the function about the height of the diver with respect to the horizontal dista

The path of a diver is given by the function f(x)=-(4)/(9)...

The path of a diver is given by the function
$$f(x)=-\frac{4}{9} x^{2}+\frac{24}{9} x+12$$
where $f(x)$ is the height (in feet) and $x$ is the horizontal distance from the end of the diving board (in feet). What is the maximum height of the diver?

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Key Concepts

Parabolic Motion

Parabolic motion refers to the trajectory that objects follow when moving under the influence of gravity, ignoring air resistance. This type of motion is typically modeled by quadratic functions, where the vertex of the parabola represents the highest point reached by the object, illustrating the practical application of quadratic optimization.

Quadratic Functions

Quadratic functions are algebraic expressions of the form ax² + bx + c where a, b, and c are constants and a is non-zero. Their graphs are parabolas that open upward or downward depending on the sign of a, making them central to solving problems involving maximum or minimum values.

Vertex of a Parabola

The vertex of a parabola is the point where the function reaches its maximum or minimum value. It can be found using the formula x = -b/(2a), and when substituted back into the function, it provides the corresponding extreme value. This concept is fundamental when solving optimization problems involving quadratic functions.

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