Determine a quadratic function f(x) for the graph given below.

A ball is thrown upward from the roof of a building with an initial velocity of 16 ft/sec. The height of the building is 50 ft. The equation for the height of the ball after t seconds is given as s(t) = -16t² + 16t + 50. Determine the time when the ball will achieve a maximum height. Also, find the maximum height.

In a certain university, the number of students who participated in the yearly fundraising activity from 2011 to 2018 can be modeled by A(x) = 3x² - 52x + 255, where x = 1 represents the year 2011. From 2018 to 2022, the number of students is represented by A(x) = 42x - 305. Graph y = A(x) for the years 2011 to 2022 and determine the year in which the number of students who participated is the least.

Justify that the polynomial f(x) = x³ + 4x - 3 has a real zero in between 0 and 1 by using the Intermediate Value Theorem.

Determine if the following function is a polynomial or not. If it is, what is its degree?

f(x) = x^1/5 - 5x⁶ + 1

Consider the function f(x) = -(x - 4)² + 6 and find the largest open interval of the domain at which the function is increasing and decreasing.

Determine the end behavior of the graph of the following polynomial function using the Leading Coefficient Test: $f(x)=-29x^{4}-3x^{2}+5x+6$