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 a ≠ 0. The graph of a quadratic function is a parabola, which opens either upwards or downwards depending on the sign of a. This lesson will guide you through the key concepts of quadratic equations, focusing on their graphs and solutions.

Graphing Quadratic Functions

To graph a quadratic function, we typically start with the standard form y = ax² + bx + c. Here are the steps to graph a quadratic function:

1. Identify the coefficients: Determine the values of a, b, and c.
2. Find the vertex: The vertex of the parabola can be found using the formula x = -b/(2a). Substitute this value back into the function to find the y-coordinate.
3. Determine the axis of symmetry: The axis of symmetry is a vertical line given by x = -b/(2a).
4. Find the y-intercept: Set x = 0 in the equation to find the y-intercept.
5. Find the x-intercepts: Solve the equation ax² + bx + c = 0 to find the x-intercepts (if they exist).
6. Plot the points and sketch the parabola: Use the vertex, axis of symmetry, intercepts, and additional points to sketch the graph.

Solving Quadratic Equations

Quadratic equations can be solved using different methods:

1. Factoring:

If the quadratic can be expressed as a product of two binomials, it can be factored. For example, x² - 5x + 6 = (x - 2)(x - 3) = 0 leads to x = 2 and x = 3.

2. Completing the Square:

This method involves rewriting the equation in the form (x - p)² = q and then solving for x. For instance, to solve x² - 6x + 5 = 0, we rewrite it as (x - 3)² = 4.

3. Quadratic Formula:

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) can be used for any quadratic equation. This method is helpful when factoring is difficult or impossible.

Analyzing Quadratic Functions

Understanding the properties of quadratic functions is crucial:

• Vertex: The highest or lowest point of the parabola, depending on the direction it opens.
• Axis of Symmetry: A line that divides the parabola into two mirror-image halves.
• Intercepts: Points where the graph intersects the axes. The y-intercept occurs when x = 0, and x-intercepts are found by solving the equation.
• Direction: If a > 0, the parabola opens upward; if a < 0, it opens downward.