Graphing Quadratic Functions in Standard Form

Overview

• Objective: Learn how to graph quadratic functions in standard form.
• Key Concept: Determine the parabola's opening direction, axis of symmetry, vertex, intercepts, and plot.

Step-by-Step Guide

1. Determine Parabola's Direction

• Standard Form: ax^2 + bx + c
• If ( a > 0 ): Parabola opens upward.
• If ( a < 0 ): Parabola opens downward.
• Example: Given ( a = 1 ) (upward).

2. Find the Axis of Symmetry

• Formula: ( x = \frac{-b}{2a} )
• Example: ( a = 1, b = -4 ) results in ( x = 2 ).
• Axis of Symmetry: Vertical line at ( x = 2 ).

3. Find the Vertex

• Vertex x-coordinate: Same as axis of symmetry.
• Vertex y-coordinate: Substitute x into the original equation.
• Example: Vertex at ( (2, -9) ).

4. Find the Y-Intercept

• Occurs: When ( x = 0 ).
• Example: Y-intercept at ( (0, -5) ).

5. Find the X-Intercepts

• Occurs: When ( y = 0 ).
• Factoring: Factor if possible or use the quadratic formula.
• Example: X-intercepts at ( (-1, 0) ) and ( (5, 0) ).
• Note: Not always easy to factor; use quadratic formula when necessary.

6. Plot the Parabola

• Points Needed: Vertex, y-intercept, x-intercepts, and symmetric points.
• Symmetric Point: Reflect the y-intercept across the axis of symmetry.
• Example: Additional point at ( (4, -5) ).
• Draw: Connect points to form the parabola.

Special Cases

Case 1: Parabola Touches X-Axis Once

• Vertex on X-Axis: Vertex is also the x-intercept.
• Example: Vertex at ( (-3, 0) ) with y-intercept ( (0, 9) ).
• Symmetric Point: ( (-3, 9) ).

Case 2: No X-Intercepts

• Occurs: When the vertex and y-intercept are above/below the x-axis and the parabola opens upward/downward respectively.
• Example: Use additional points close to the axis of symmetry for plotting.

Practical Tips

• Five Points: Minimum recommended for plotting a decent parabola.
• Accuracy: More points can improve accuracy.
• Calculator Use: Necessary when the discriminant isn't a perfect square.

Conclusion

• Follow steps for determining direction, axis of symmetry, vertex, and intercepts.
• Use plotting techniques and symmetry for graph accuracy.
• Helpful resources and examples in provided links or videos.

Note: Practice with different examples to reinforce your understanding of graphing quadratic functions.