Overview
This lesson explains how to graph quadratic functions in standard form by identifying key features such as the direction, axis of symmetry, vertex, and intercepts.
Determining the Parabola's Direction
- For a quadratic function (y = ax^2 + bx + c), if (a > 0), the parabola opens upward; if (a < 0), it opens downward.
Finding the Axis of Symmetry
- The axis of symmetry is given by (x = \frac{-b}{2a}).
- The axis of symmetry divides the parabola into two equal halves.
Finding the Vertex
- The vertex lies on the axis of symmetry; its x-coordinate is (x = \frac{-b}{2a}).
- Substitute the x-coordinate into the function to find the y-coordinate of the vertex.
Identifying Intercepts
- Y-intercept: Set (x = 0) and solve for (y) ((y = c)).
- X-intercepts: Set (y = 0) and solve for (x) (may require factoring or the quadratic formula).
Plotting Key Points and Symmetry
- Key points include the vertex, y-intercept, x-intercepts, and points symmetric to the y-intercept and other chosen points across the axis of symmetry.
- Use at least five points for an accurate graph.
Special Cases
- If the discriminant is zero, the vertex lies on the x-axis, and the parabola touches the x-axis at one point.
- If the discriminant is negative, there are no real x-intercepts, and the parabola lies entirely above or below the x-axis.
Strategies for Difficult X-Intercepts
- If factoring or the quadratic formula is hard, choose additional x-values near the axis of symmetry and find their corresponding y-values.
- Reflect these points across the axis of symmetry for additional accuracy.
Key Terms & Definitions
- Quadratic Function — A function of the form (y = ax^2 + bx + c).
- Axis of Symmetry — Vertical line dividing the parabola into two equal halves, (x = \frac{-b}{2a}).
- Vertex — The highest or lowest point on the parabola, located on the axis of symmetry.
- Y-intercept — The point where the parabola crosses the y-axis ((x=0)).
- X-intercepts — Points where the parabola crosses the x-axis ((y=0)).
- Discriminant — (b^2 - 4ac), determines the nature of the roots (real or complex).
Action Items / Next Steps
- Practice graphing quadratics by identifying direction, axis of symmetry, vertex, and intercepts for given functions.
- Try the sample problems and pause on suggested exercises in the lesson.