Overview
This lesson covers the vertex form of quadratic functions, focusing on interpreting and graphing parabolas with parameters (p) and (q), and analyzing the effects of shifting graphs horizontally and vertically.
Basic Quadratic Function (y = x^2)
- The function (y = x^2) is a basic quadratic with a parabolic graph.
- Its vertex is at (0, 0), minimum value is 0 at (x = 0), and there is no maximum value.
- The axis of symmetry is (x = 0).
- Both (x)- and (y)-intercepts are at (0, 0).
- Domain: all real (x); Range: (y \geq 0).
Horizontal Shifts ((y = (x - p)^2))
- Replacing (x) with (x - p) shifts the graph horizontally.
- If (p > 0), the graph shifts right (p) units; if (p < 0), it shifts left (|p|) units.
- The new vertex is at ((p, 0)), axis of symmetry is (x = p).
- The minimum remains 0, occurring at (x = p).
- Domain remains all real numbers; Range remains (y \geq 0).
Vertical Shifts ((y = x^2 + q))
- Adding (q) to (x^2) shifts the graph vertically.
- If (q > 0), the graph moves up (q) units; if (q < 0), it shifts down (|q|) units.
- New vertex at (0, (q)); minimum is (q) at (x = 0).
- Axis of symmetry is (x = 0).
- Domain: all real numbers; Range: (y \geq q).
General Vertex Form (y = (x - p)^2 + q)
- The vertex is at ((p, q)).
- The axis of symmetry is (x = p).
- Minimum (or maximum if the parabola opens down) is at (y = q) when (x = p).
- Domain: all real numbers; Range: (y \geq q) (if parabola opens up).
- The p-value is always opposite sign to the number in the bracket (e.g., (x + 2) means (p = -2)).
Example: Describing Transformations
- (y = (x + 2)^2 - 4): horizontal shift 2 units left, vertical shift 4 units down.
- Vertex: ((-2, -4)). Axis of symmetry: (x = -2).
- Domain: all real numbers. Range: (y \geq -4).
- X-intercepts found by setting (y = 0) and solving.
Key Terms & Definitions
- Quadratic Function — A polynomial function of degree 2; graph is a parabola.
- Vertex — The peak or lowest point on a parabola.
- Axis of Symmetry — Vertical line dividing the parabola into two mirror halves ((x = p)).
- Domain — Set of all possible (x) values.
- Range — Set of all possible (y) values.
- p-value — Controls horizontal shift; vertex x-coordinate.
- q-value — Controls vertical shift; vertex y-coordinate.
- Parabola — The curved graph of a quadratic function.
Action Items / Next Steps
- Complete all assigned questions and exercises for practice.
- Fill in any unfinished tables for different quadratic forms.
- Sketch graphs of quadratic functions given different (p) and (q) values.