Question

The function y=-0.2x^2+1.9x models the path of a kicked soccer ball.The height is y,the distance is x, and the units are meters.How high does the soccer ball go?

Answer 1

The highest height the ball achieves is 4.5125 meters.

Explanation:

Vertex of a quadratic function:

Suppose we have a quadratic function in the following format:

`f(x) = ax^(2) + bx + c`

It's vertex is the point
`(x_(v), y_(v))`

In which

`x_(v) = -(b)/(2a)`

`y_(v) = -(\Delta)/(4a)`

Where

`\Delta = b^2-4ac`

If a<0, the vertex is a maximum point, that is, the maximum value happens at
`x_(v)`, and it's value is
`y_(v)`.

In this question:

Quadratic function
`y = -0.2x^2 + 1.9x`, which has
`a = -0.2, b = 1.9`.

The maximum height of the ball is
`y_(v)`.

Then

`\Delta = b^2-4ac = (1.9)^2 - 4*(-0.2)(0) = 3.61`

`y_(v) = -(3.61)/(4(-0.2)) = 4.5125`

The highest height the ball achieves is 4.5125 meters.