Vertex Form Exploration

The equation of a quadratic function can be written in the form f(x) = a(x - h)2 + k. Turn on the trace feature of the red function first. Use the applet below to explore how changing the values of a, h, and k affect the parent graph f(x) = x2.

What happens to the function when you change "a" to be negative?

What happens the function when you change "h"?

What happens the function when you change "k"?

Use your new information about vertex form to find the quadratic function that best matches the picture of the St. Louis Arch.

What quadratic equation best models the St. Louis Arch? Please write your equation in vertex form.

The applet below allows for you to reflect the parabola across the x-axis. Use the applet to develop a conjecture about how this reflection affects the ordered pairs that lie on the parabola. The parabola is shown in red and its reflective image is shown in blue.
The applet below lets you experiment with a vertical translation of a parabola. The parabola is also shown after it is reflected over the x-axis.

What relationship exists between the value of k of the original parabola and the distance between the original parabola and its reflective image?