Quadratic Graphs Lecture Notes

Introduction to Quadratic Graphs

• Quadratic graphs represent quadratic equations.
• Characteristics of quadratic equations:
  • Include an x² term.
  • Do not have higher powers of x (e.g., x³, x⁴).

Key Features of Quadratic Graphs

• Shape: Smooth, curved shape.
• Line of Symmetry: Present in the middle, making the graph symmetrical on both sides.
• Direction of Curve:
  • If x² term is positive, the curve faces upwards (smiley face).
  • If x² term is negative, the curve faces downwards (unhappy face).

Simple Quadratic Graphs

• Equation: y = x²
  • Graph crosses the axis at (0, 0).
• Equation: y = -x²
  • Curve is downward due to the negative x² term.

Impact of Changing the Equation

• Changes in the equation affect the graph's shape and position.
• Examples:
  • y = 2x² makes the curve narrower.
  • y = 2x² - 3 moves the curve downwards.
  • y = 2x² - 5x - 3 moves the curve further down and to the right.

Conclusion

• Understanding changes in the equation helps predict changes in the graph's appearance.
• No need to understand the exact mechanics of each change, but recognize that changes in numbers affect the graph's shape.