Overview
This lecture introduces quadratic graphs, explains their key features, and describes how changes in the equation affect the graph's shape and position.
What Makes a Quadratic Graph
- Quadratic graphs represent equations containing an x squared (x²) term, but no higher powers like x³ or x⁴.
- The general form of a quadratic equation is y = ax² + bx + c.
Key Features of Quadratic Graphs
- Quadratic graphs always have a smooth, curved (parabola) shape.
- There is always a line of symmetry down the middle, making the left and right sides mirror images.
- If the x² term is positive, the curve faces upwards (like a smiley face).
- If the x² term is negative, the curve faces downwards (like an unhappy face).
- The simplest quadratic graph, y = x², crosses the axis only at (0, 0).
- The graph y = -x² curves downwards and also crosses at (0, 0).
Effects of Changing Equation Terms
- Increasing the coefficient of x² (e.g., y = 2x²) makes the curve narrower.
- Subtracting a number (e.g., y = 2x² - 3) shifts the graph down.
- Adding or subtracting bx terms (e.g., y = 2x² - 5x - 3) can shift the graph left or right and further up or down.
Key Terms & Definitions
- Quadratic Equation — an equation containing an x² term and no higher powers.
- Parabola — the U-shaped curve of a quadratic graph.
- Line of Symmetry — a vertical line dividing the parabola into two mirror-image halves.
- Coefficient — the number in front of a variable (e.g., the 2 in 2x²).
Action Items / Next Steps
- Review your textbook's section on quadratic graphs and their features.