# Parabolas and Quadratic Equations Study Guide

This comprehensive guide covers essential concepts related to parabolas and quadratic equations, focusing on different forms of quadratic expressions, conversion methods, and solving techniques.

**Definition**: A parabola is a U-shaped curve that can be represented by a quadratic equation.

## Forms of Quadratic Expressions

Quadratic expressions can be written in three main forms:

- Standard form: ax² + bx + c
- Vertex form: a(x-h)² + k
- Intercept form: a(x-p)(x-q)

**Highlight**: The vertex form is particularly useful for identifying the parabola's turning point and axis of symmetry.

## Vertex of a Parabola

The vertex of a parabola can be found using two methods:

- x-coordinate: -b/(2a) (also the axis of symmetry)
- (h,k) in vertex form

**Example**: If a > 0, the vertex is a minimum point; if a < 0, the vertex is a maximum point.

## Finding x-intercepts

X-intercepts can be determined by:

- Factoring the quadratic expression
- Using the quadratic formula or completing the square

## Completing the Square

**Completing the square** is a method used to convert from standard form to vertex form and solve quadratic equations. The process involves the following steps:

- Add (b/2a)² to both sides of the equation
- Factor the left side and simplify the right side
- Solve for intercepts by taking the square root of both sides and subtracting the extra term

**Vocabulary**: **Completing the square** is a technique used to rewrite a quadratic expression as a perfect square trinomial plus a constant.

## Converting to Vertex Form

To convert a quadratic expression from standard form to vertex form:

- Rewrite as: a(x² + (b/a)x) + c
- Complete the square: a(x² + (b/a)x + (b/2a)²) + c - a(b/2a)²
- Factor: a(x + b/2a)² + (c - b²/4a)

**Highlight**: The vertex form a(x-h)² + k is derived from completing the square.

## The Quadratic Formula

The quadratic formula is used to solve quadratic equations:

x = (-b ± √(b²-4ac)) / (2a)

**Definition**: The **discriminant** is the expression under the square root in the quadratic formula: b²-4ac.

## Understanding the Discriminant

The discriminant helps determine the nature of a quadratic equation's roots:

- Positive: Two real solutions
- Perfect square: Two rational solutions
- Zero: One real solution (the vertex)
- Negative: No real solutions (two imaginary solutions)

**Example**: For the equation x² + 4x + 4 = 0, the discriminant is 4² - 4(1)(4) = 0, indicating one real solution.

## Systems of Quadratic Equations

When solving systems involving quadratic equations:

- Graph to check for intersections
- Only consider real solutions

## Quadratic Inequalities

When solving quadratic inequalities:

- Choose a test point not on the parabola to determine which region to shade
- For inequalities with two variables, perform a point test for both equations and shade the shared regions

**Highlight**: The point test is crucial for determining the solution region of quadratic inequalities.

This study guide provides a comprehensive overview of parabolas and quadratic equations, covering essential concepts and techniques for solving various problems related to these topics.