Overview
The lesson covers how to graph quadratic functions in standard form, focusing on identifying key features such as direction, axis of symmetry, vertex, and intercepts.
Determining Parabola Direction
- The standard form of a quadratic is y = ax² + bx + c.
- If a > 0, the parabola opens upward; if a < 0, it opens downward.
Calculating Key Features
- The axis of symmetry formula is x = -b / (2a).
- The vertex lies on the axis of symmetry: the x-coordinate is x = -b/(2a).
- Find the vertex’s y-coordinate by substituting the vertex’s x value into the original function.
- The y-intercept is found by setting x = 0 in the quadratic; it equals the constant c.
- X-intercepts are found by setting y = 0 and solving for x, using factoring or the quadratic formula when possible.
Plotting Points & Symmetry
- Plot the axis of symmetry as a vertical line.
- Plot vertex, y-intercept, and x-intercepts (if they exist).
- For accuracy, add points symmetric to the y-intercept and other plotted points across the axis of symmetry.
Special Cases
- When the vertex’s y-coordinate is 0, the vertex is also the x-intercept (one real solution).
- If the discriminant (b² - 4ac) is negative, the quadratic has no real x-intercepts and the parabola does not cross the x-axis.
- With only three main points, add extra symmetric points closer to the axis of symmetry for a more accurate graph.
Key Terms & Definitions
- Quadratic Function — a function in the form y = ax² + bx + c.
- Parabola — the graph of a quadratic function.
- Axis of Symmetry — a vertical line dividing the parabola into two equal halves, x = -b/(2a).
- Vertex — the maximum or minimum point on the parabola.
- Y-Intercept — the point where the parabola crosses the y-axis (x = 0).
- X-Intercept(s) — point(s) where the parabola crosses the x-axis (y = 0).
- Discriminant — b² - 4ac, determines the nature of the roots.
Action Items / Next Steps
- Practice graphing quadratics with different values of a, b, and c.
- Try plotting at least five points for each parabola, including symmetric points.
- Review special cases: single and no real solutions for x-intercepts.