Suppose a goalie kicks a soccer ball. The ball travels in a parabolic path from point (0,0) to (57,0). Consider a quadratic function in vertex form for the path of the ball. Which values can you determine? What values are you unable to determine? Explain.

The correct answer is: we will be able to determine the x-coordinate(h) of the vertex only. The value of y-coordinate(k) and “a” of the equation of the path of the ball

a)
Let’s say that the equation of the path of the ball is y  = a(x – h)² + k
The ball travels from (0,0) to (57,0).
Put (0,0) in the equation y  = a(x – h)² + k

0 = a(0-h)² + k

k = -ah²    …….(1)
Put (57,0) in the equation y  = a(x – h)² + k

0 = a(57-h)² + k

a(3249 + h² - 114h) - ah² = 0     ( from equation 1)

3249a +ah² - 114ha - ah² = 0

3249a = 114ha

2h = 57

h = 28.5
Final Answer:
Hence, we will be able to determine the x-coordinate(h) of the vertex only. The value of y-coordinate(k) and “a” of the equation of the path of the ball.

A polynomial function is referred to as quadratic if it has one or more variables and a variable with a maximum exponent of two. It is sometimes referred to as the polynomial of degree 2 since the greatest degree term in a quadratic function is of the second degree.
The locations whose coordinates are of the form are connected by the parent quadratic function, which has the form f(x) = x2 (number, number2). The parent quadratic function joins the places whose coordinates have the form f(x) = x2 (number, number2). This function, which generally has the form f(x) = a (x - h)2 + k, can be transformed to take the form f(x) = ax2 + bx + c.