Teacher's Guide

Lesson Objectives

• Understand the relationship between coefficients a, b, c and the graph of a quadratic function.
• Visualize how the discriminant (b² - 4ac) determines the number and type of roots.
• Connect the algebraic formula to geometric features (vertex, axis of symmetry).

Guided Activity Ideas

1. The "A" Effect

Goal: Explore concavity and width.

1. Ask students to set b = 0 and c = 0.
2. Have them vary a from positive to negative.
3. Discussion: What happens when a = 0? Is it still a quadratic?

2. Hunting for Roots

Goal: Understand the discriminant.

• Create a quadratic with 2 real roots (e.g., y = x² - 4).
• Create one with 1 real root (hint: vertex on x-axis).
• Create one with no real roots.
• Have students observe the value under the square root in the formula panel.

Discussion Questions

• Why does the graph always pass through (0, c)?
• How does changing b affect the position of the vertex? (It moves along a specific curve!)
• Is it possible to have a quadratic with 3 roots? Why or why not?