Solving and Graphing a Quadratic Equation

Introduction to Quadratic Equations

• A quadratic equation is a polynomial equation of the form ax² + bx + c = 0, where a, b, and c are constants and x represents the variable.
• The graph of a quadratic equation is a parabola.

Solving Quadratic Equations

Methods to Solve Quadratic Equations

1. Factoring
   • Express the quadratic in the form (x - p)(x - q) = 0
   • Solutions are x = p and x = q
2. Completing the Square
   • Rewriting the equation in the form (x - h)² = k
   • Solve for x using the square root method
3. Quadratic Formula
   • x = (-b ± √(b² - 4ac)) / 2a
   • Useful for any form of quadratic equations

Finding Key Features of the Parabola

• Axis of Symmetry: x = -b / 2a
• Vertex: The highest or lowest point on the graph
• X-Intercepts: Points where the graph crosses the x-axis
• Y-Intercept: The value of y when x = 0

Graphing Quadratic Equations

Steps to Graph

1. Find the Axis of Symmetry
   • Use x = -b / 2a
2. Calculate the Vertex
   • Vertex (h, k) where h is the axis of symmetry
   • Substitute h into the equation to find k
3. Determine the Y-Intercept
   • Substitute x = 0 in the equation to find the y-intercept
4. Find Additional Points
   • Choose x-values and solve for y to plot additional points
5. Draw the Parabola
   • Plot the vertex, axis of symmetry, and additional points
   • Sketch a smooth curve through these points

Types of Solutions

• Two Solutions: The parabola crosses the x-axis at two points
• One Solution: The parabola touches the x-axis at a single point (vertex)
• No Real Solutions: The parabola does not cross the x-axis

Conclusion

• Understanding how to solve and graph quadratic equations is foundational in algebra.
• Practice different methods and examples to gain proficiency.