Welcome to Math Mastery PRO!

Understanding Linear Equations (y = mx + b)

A linear equation represents a straight line on a graph. The standard form is y = mx + b, where:

• y is the dependent variable (output).
• x is the independent variable (input).
• m is the slope, indicating the steepness and direction of the line (rise over run).
• b is the y-intercept, the point where the line crosses the y-axis (when x=0).

Understanding these components allows you to graph the line and predict its behavior.

Interactive Playground: Explore y = mx + b

Current Equation: y = 1.0x + 0.0

Understanding Quadratic Equations (ax² + bx + c = 0)

Quadratic equations describe parabolas, which are U-shaped curves. The standard form is ax² + bx + c = 0. Key concepts include finding the roots (where the parabola crosses the x-axis) using the quadratic formula, factoring, or completing the square. The coefficient 'a' determines if the parabola opens upwards (a > 0) or downwards (a < 0).

Interactive Playground: Explore Parabolas

Current Equation: y = 1.0x² + 0.0x + 0.0

Understanding the Pythagorean Theorem (a² + b² = c²)

The Pythagorean Theorem is a fundamental principle in Euclidean geometry stating that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides (legs).

The formula is expressed as: a² + b² = c², where:

• a and b are the lengths of the legs of the right triangle.
• c is the length of the hypotenuse.

This theorem is incredibly useful for finding unknown side lengths in right triangles and has applications in various fields like construction, navigation, and physics.

Interactive Playground: Visualize a² + b² = c²

Calculated Hypotenuse (c): 5.00

(3.0² + 4.0² = 5.0²) => 9.0 + 16.0 = 25.0

Understanding Circles

A circle is a shape consisting of all points in a plane that are a given distance (the radius) from a given point (the center). Key properties include:

• Radius (r): Distance from the center to any point on the circle.
• Diameter (d): Distance across the circle through the center (d = 2r).
• Circumference (C): Distance around the circle (C = 2πr or C = πd).
• Area (A): Space enclosed by the circle (A = πr²).

Interactive Playground: Circle Properties

Circumference: 31.42 | Area: 78.54

Introduction to Derivatives

Calculus is the study of change. A derivative measures the instantaneous rate of change of a function, or the slope of the tangent line to the function's graph at a specific point.

One of the fundamental rules for finding derivatives is the Power Rule. For a function of the form f(x) = xⁿ, where n is a constant, its derivative, denoted as f'(x) or dy/dx, is given by:

f'(x) = n * xⁿ⁻¹

For example, if f(x) = x³ (so n=3), then f'(x) = 3 * x³⁻¹ = 3x².

Derivatives have wide applications in physics (velocity, acceleration), economics (marginal cost/revenue), optimization problems, and more.

Interactive Playground: Power Rule Explorer

Function: f(x) = x2

Derivative: f'(x) = 2x1

Introduction to Integrals

Integration is often thought of as the reverse process of differentiation. A definite integral, represented as ∫[a, b] f(x) dx, calculates the net signed area between the function f(x) and the x-axis, from x=a to x=b. An indefinite integral, ∫ f(x) dx, finds the family of functions (antiderivatives) whose derivative is f(x). The power rule for integration states that ∫ xⁿ dx = (xⁿ⁺¹ / (n+1)) + C, for n ≠ -1.

Interactive Playground: Area Under Curve

Approximate Area: Calculating...