Imaginary and Complex Numbers

This page delves into imaginary and complex numbers, providing definitions and examples of operations involving these numbers. It covers the fundamental concept that i² = -1 and how this applies to various calculations.

Vocabulary: Complex numbers are numbers in the form a + bi, where a is the real part and b is the imaginary part.

The page includes examples of simplifying expressions with imaginary numbers and performing operations with complex numbers. It also introduces the concepts of rational and irrational numbers.

Example: The page demonstrates how to simplify expressions like (5i)⁴, breaking it down step-by-step to arrive at the solution 625.

Highlight: When working with powers of i, remember that the pattern repeats every four powers: i¹ = i, i² = -1, i³ = -i, i⁴ = 1.

Equation of Circles and Vertex Form of Parabolas

This final page covers two important topics: the equation of circles and the vertex form of parabolas. It provides the general form of a circle equation and explains how to identify the center and radius from the equation.

Definition: The general form of a circle equation is (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.

The page also discusses the vertex form of parabolas for both vertical and horizontal orientations. It explains how to interpret the coefficients in these forms to determine the direction of opening and the location of the vertex.

Example: For the parabola equation (x + 1)² = -8(y - 4), the page demonstrates how to identify the vertex, axis of symmetry, directrix, and focus.

Highlight: Understanding the vertex form of parabolas is crucial for quickly identifying key features of the graph, such as the vertex, direction of opening, and axis of symmetry.

Quadratic Function Exam Review

This page introduces key concepts for solving quadratic equations and graphing quadratic functions. It covers the fundamental theorem of quadratics, rules for solving equations, and methods for graphing.

Definition: The fundamental theorem of quadratics states that you have as many answers as the greatest roots.

Example: For solving quadratic equations, the page provides examples such as 3k² = 8k - 4, demonstrating the step-by-step process to find solutions.

The page also explains how to graph quadratic equations, including finding the axis of symmetry, vertex, y-intercept, and x-intercepts. It emphasizes the importance of understanding whether the parabola opens upward or downward based on the sign of the leading coefficient.

Highlight: When graphing quadratic equations, pay attention to the sign of the leading coefficient to determine if the parabola opens upward (positive) or downward (negative).