Overview
This lecture covers methods for graphing quadratic functions using transformations without creating a full table of values, focusing on the effects of modifying parameters in the standard form.
Quadratic Function Transformations
- The basic quadratic function is y = x², which graphs as a parabola.
- Multiplying by a constant "a" (y = a·x²) changes the parabola's width and orientation.
- If |a| > 1, the graph becomes narrower (vertical stretch); if 0 < |a| < 1, it becomes wider (vertical compression).
- If a > 0, the parabola opens upward; if a < 0, it opens downward (reflection over x-axis).
- Adding or subtracting a value inside the squared term (y = a(x – p)²) shifts the graph left or right.
- If p > 0, the graph shifts right by p units; if p < 0, it shifts left by |p| units.
- Adding a constant outside the squared term (y = a(x – p)² + q) shifts the graph vertically.
- If q > 0, shift up by q units; if q < 0, shift down by |q| units.
Graphing in Standard Form
- The standard form is y = a(x – p)² + q, where:
- a determines vertical stretch/compression and opening direction.
- p shifts the graph horizontally.
- q shifts the graph vertically.
- The vertex of the graph is at the point (p, q).
- The axis of symmetry is the vertical line x = p.
- Use symmetry to find additional points: for each point (x, y), there is a symmetric point across the axis.
- The minimum or maximum value is at the vertex: minimum if a > 0, maximum if a < 0.
- The domain of a quadratic is all real numbers; the range starts at the vertex’s y-value and goes up (if opening up) or down (if opening down).
Finding the Equation from Properties
- To find a standard form equation from a vertex and another point (like an x-intercept):
- Substitute the vertex into (p, q) in y = a(x – p)² + q.
- Plug the known point into x and y, then solve for a.
- Write the full equation using the found values.
Key Terms & Definitions
- Vertex — The highest or lowest point of the parabola, located at (p, q).
- Axis of Symmetry — Vertical line x = p dividing the parabola into two mirror images.
- Vertical Stretch/Compression — Changes in the parabola's width due to coefficient a.
- Translation — Shifting the entire graph left/right (by p) or up/down (by q).
- Standard Form — The equation y = a(x – p)² + q for a quadratic function.
Action Items / Next Steps
- Practice graphing quadratic functions in standard form using transformations.
- Given vertex and an additional point, determine the equation of the parabola.
- Complete assigned homework problems on quadratic transformations.