Transformations in Baseball

A sports science consultant was hired to analyze the throwing arm of our star outfielder. It was determined that the ball was thrown a distance of 100 feet, at a peak height of 40 feet. Find a formula (a function ) that gives the height of the ball as a function of the distance from the outfielder, by transforming the parent function (shown in dashed blue).
A ball thrown on Planet Earth follows a parabolic trajectory due to gravity. That is, the path of the ball is a parabola. The parent function for a parabola is , but clearly this by itself does not properly describe the path of the ball. To turn a parent function into something useful for modeling real life, we can apply transformations. Transformations stretch and shift a graph to make the parent function's shape "fit" the real-life scenario. Adjust the sliders to transform parent function into a model for the throw, which we'll call .