Question bank: Second Degree Function: Graph and Table

In a dart throwing competition, the trajectory of the dart can be modeled by the quadratic function f(x) = -2x² + 12x + 5, where x is the horizontal distance traveled by the dart in meters and f(x) is the height of the dart above the ground in meters. Based on this model, at what point does the dart reach its maximum height? And what is that height? Additionally, at what distance from the starting point does the dart touch the ground?

Which of the options below correctly represents the graph of the quadratic polynomial function f(x) = ax² + bx + c, where a > 0?

Consider the function f(x) = x^2 - 4x - 5. Determine the value of f(3) and graphically represent the function in the interval of x belonging to [-2, 6] using a computational tool or by describing the step-by-step process for constructing the graph.

Imagine that a marble producer company received a special order to create a line of marbles with diameters that vary according to a quadratic function. The function that relates the diameter D (in millimeters) of the marbles with the time t (in weeks) since the beginning of production is given by D(t) = -2t² + 12t + 5.

A theme park is planning the opening of a new roller coaster with a different and intriguing shape. The main trajectory of this roller coaster is in the form of a parabola, represented by the quadratic function f(x) = ax² + bx + c on the Cartesian plane. It is known that the roller coaster has a steep ascent starting from point (1, 10) and, after reaching its highest point at (3, 20), it begins to descend to its lowest point at (5, 10). Based on this information, solve the following items: (a) Determine the quadratic function f(x) that represents the trajectory of the roller coaster. (b) Sketch the graph of the function f(x) on the Cartesian plane, indicating the points of interest. (c) Use the graph to identify the type of parabola (facing upwards or downwards) and the proportionality relationship between the x and y coordinates of the parabola.