Graphing Quadratic Functions

During basketball practice, motion-tracking systems can record a ball’s horizontal distance and height. We can model its trajectory with a quadratic function.

To graph the trajectory, we rewrite the quadratic function in vertex form by completing the square. Then, we transform the parent parabola.

First, we factor out the leading coefficient from the x-terms. Then, we square half the coefficient of x and add it inside the parentheses. We adjust the outside expression so the function stays the same.

In this form, (h, k) gives the vertex. Suppose a equals negative 0.1, h equals 4, and k equals 3.7. To graph this function, we start with the parent parabola.

Since a is negative, the parabola opens downward. Its magnitude is less than 1, so the parabola is wider than the parent parabola. A positive h shifts the graph right, while a positive k shifts it upward. The transformed graph has a vertex at (4, 3.7), representing the ball’s maximum height and the horizontal position where it reaches that height.

So, graphing the quadratic function helps us visualize the ball’s path and understand its key features.