Question

Cathy hit a golf ball 180 yards down the fairway. If the ball reached a maximum height of 25 yards, what is the equation(in graphing form) for the height of the golf ball versus the horizontal distance it has traveled? Assume a parabolic path.

Answer 1

See below.

Explanation:

Let the x axis be the horizontal distance and the y axis be the vertical distance.

We know the ball travelled 180 yards. If we construct a quadratic equation with roots at `x=0` the starting point and `x=180` the final distance, then we just have to make the maximum value of the function `y=25`.

We know the vertex of a parabola has an axis of symmetry midpoint of the roots. We use `(180-0)/2=90` to find this.

Using vertex form of a quadratic:

`y=a(x-h)^2+k`

Where:

`bbacolor(white)(888)` is the coefficient of `x^2`.

`bbhcolor(white)(888)` is the axis of symmetry.

`bbkcolor(white)(888)` is max/min value.

Plugging in what we know, and using one of the roots:

`y=a(0-(90))^2+25`

we now solve for `bba`

First we need a value for `y`. We know at `x=0` the ball is sill on the ground, so we can say at `x=0=>y=0`

So:

`a(0-(90))^2+25=0`

`90^2a+25=0`

`90^2a=-25`

`a=-25/90^2=-1/324`

`:.`

`y=-1/324(x-90)^2+25`

This is the required equation. We can expand this if we need to, so as to have the form `ax^2+bx+c`.

GRAPH: