Overview
This lecture explains how to graph quadratic equations, interpret their key features, and find their equations using different forms.
Quadratic Equations and Their Graphs
- A quadratic equation has the form f(x) = ax² + bx + c, where a ≠0.
- The graph of f(x) = x² is a parabola that opens upward.
- The value of "a" affects the parabola's width and orientation: larger |a| squashes or stretches, negative "a" flips it.
- Many real-world objects, like satellite dishes, have parabolic shapes.
Standard and Vertex Forms
- Standard form: f(x) = ax² + bx + c.
- Vertex form: f(x) = a(x–h)² + k, where h = –b/(2a) and k = f(h).
- h determines the horizontal shift; k determines the vertical shift of the parabola.
- The vertex (h, k) is the lowest or highest point of the parabola.
- The axis of symmetry is the vertical line x = h.
Example: Graphing a Quadratic
- For f(x) = 2x² – 12x + 16: a = 2, b = –12, c = 16.
- Calculate h: h = –b/(2a) = –(–12)/(2×2) = 3.
- Calculate k: k = f(3) = 2(3)² – 12×3 + 16 = 2.
- The vertex is at (3, 2) and the axis of symmetry is x = 3.
- The graph opens upward and is narrower than f(x) = x².
Finding an Equation from a Graph
- If the vertex (h, k) and another point are known, use f(x) = a(x–h)² + k.
- Substitute the known point to solve for "a".
- Confirm symmetry and orientation with the sign of "a" and by plotting extra points.
Key Terms & Definitions
- Quadratic Equation — An equation of the form ax² + bx + c with a ≠0.
- Parabola — The U-shaped graph of a quadratic function.
- Vertex — The maximum or minimum point of a parabola, at (h, k).
- Axis of Symmetry — The vertical line x = h that divides the parabola symmetrically.
- Standard Form — f(x) = ax² + bx + c.
- Vertex Form — f(x) = a(x–h)² + k.
Action Items / Next Steps
- Practice graphing quadratics and identifying their vertex, axis, and direction.
- Try converting from standard to vertex form and vice versa for different equations.
- Use given graphs to find possible quadratic equations by identifying vertex and another point.