Here's the shape of y = x². It's a U-shaped curve called a parabola, with its lowest point (the vertex) at the origin (0, 0). The graph is symmetric: x = 2 and x = -2 both give y = 4, and x = 3 and x = -3 both give y = 9. Because squaring never produces a negative number, the curve never dips below the x-axis, and it gets steeper the farther you move from zero.
A quick table to plot by hand:
- x = -3, y = 9: a point on the left arm
- x = -2, y = 4: left side, closer to the vertex
- x = -1, y = 1: one step from the bottom
- x = 0, y = 0: the vertex, where the curve bottoms out
- x = 1, y = 1 and x = 2, y = 4, x = 3, y = 9: the mirror image on the right
Plotting those seven points and connecting them smoothly gives the familiar U. A good follow-up for her is to change the formula to y = x² + 3 and ask what moves: the whole curve shifts up three units, with the vertex now at (0, 3).
If you'd like to see an interactive version, a graphing calculator site such as Desmos will draw it instantly and let her drag the sliders.
