AYRUN(Always Run)

Unit 9: Series Geometric Series A geometric series adds numbers that multiply by the same amount each time. Example: 1 + 2 + 4 + 8 + 16... Show Geometric Series Geometric Series (with Summation Notation) We can write a geometric series like this: Σ (from n = 0 to N) of arⁿ This is called summation notation . Arithmetic Series An arithmetic series adds numbers that increase by the same amount each time. Example: 2 + 4 + 6 + 8 + 10... Show Arithmetic Series The Binomial Theorem The binomial theorem helps us expand expressions like (a + b)ⁿ. It's like a recipe to figure out all the terms! Quiz: Test Your Understanding If we st...

Unit 7: Matrices Introduction to Matrices A matrix is a rectangular array of numbers organized in rows and columns. Using Matrices to Represent Data Matrices can store and manipulate data, such as images, graphs, or systems of equations. Multiplying Matrices by Scalars Each element of the matrix is multiplied by the scalar value. Adding and Subtracting Matrices Add/subtract corresponding elements of two matrices of the same dimensions. Using Matrices to Transform the Plane 2x2 matrices can scale, rotate, shear, and reflect shapes on the plane. Show Plane Transformation Transforming 3D and 4D Vectors with Matrices Higher-dimensional matrices can transform 3D/4D vectors (used in com...

Unit 10: Limits and Continuity 🎓 Defining Limits and Using Limit Notation 👉 Limit means what a function is "getting close to" as x gets close to a certain value. Example notation: lim x → 2 f(x) = 5 means "as x approaches 2, f(x) approaches 5." Estimating Limits from Graphs We can look at a graph to guess what the limit is! Let's try with this example: Show Limit Graph Exploring Types of Discontinuities Sometimes a function "jumps" or has a "hole." These are called discontinuities . Show Discontinuity Example Defining Continuity at a Point A function is continuous at x = a if: ✅ f(a) is defined ✅ lim x → a f(x) exists ✅ lim x...

Unit 8: Probability and Combinatorics Venn Diagrams and the Addition Rule Venn Diagrams help us see how different groups overlap. Addition Rule: If A and B are two events, then: P(A or B) = P(A) + P(B) - P(A and B) Show Venn Diagram Multiplication Rule for Probabilities If two events are independent, P(A and B) = P(A) × P(B). Permutations Permutations = number of ways to arrange things in order. Example: How many ways can 3 toys be arranged on a shelf? Combinations Combinations = number of ways to choose things when order does NOT matter. Example: How many ways to pick 2 toys from 5? Probability Using Combinatorics Use combinations to count outcomes and calculate prob...

Unit 3: Complex Numbers The Complex Plane Complex numbers are represented as points (or vectors) on a plane where the x-axis is the real part and the y-axis is the imaginary part. Show Complex Plane Modulus and Argument The modulus of a complex number z = a + bi is √(a² + b²). The argument (angle) is the angle with the real axis. Show Polar Representation Polar Form of Complex Numbers z = r (cos θ + i sin θ) = r·e^(iθ) Graphically Multiplying Complex Numbers Multiplying complex numbers adds their angles and multiplies their moduli. Complex Conjugates The conjugate of z = a + bi is a - bi. It reflects the point across the real axis. Distance and Midpoint of Complex Num...

Unit 5: Conic Sections Introduction to Conic Sections Conic sections include circles, ellipses, parabolas, and hyperbolas, formed by slicing a cone at various angles. Center and Radii of an Ellipse The standard form of an ellipse: (x - h)²/a² + (y - k)²/b² = 1. Center at (h, k), radii a and b. Show Ellipse Foci of an Ellipse The distance to each focus is c = √(a² - b²), located along the major axis from the center. Introduction to Hyperbolas Standard form: (x - h)²/a² - (y - k)²/b² = 1 (horizontal), or flipped for vertical hyperbola. Show Hyperbola Foci of a Hyperbola Distance to each focus is c = √(a² + b²), further from the center than the vertices. Hyperbolas Not C...

Unit 1: Composite and Inverse Functions Composing Functions When composing functions, the output of one function becomes the input of another. Example: f(g(x)). Visualize Example Modeling with Composite Functions Composite functions are used to model real-world problems with multiple steps. For example, converting units and then calculating costs. Example: Celsius → Fahrenheit → Energy cost Invertible Functions A function is invertible if its graph passes the horizontal line test — each output is mapped from exactly one input. Show Example Graph Inverse Functions in Graphs and Tables Inverse functions reflect across the line y = x. On a graph, f(x) and its inverse are mirror images. Visualize Reflection ...

Unit 4: Rational Functions Reducing Rational Expressions to Lowest Terms To reduce a rational expression, factor numerator and denominator and cancel common terms. End Behavior of Rational Functions The degrees of the numerator and denominator determine the end behavior (horizontal asymptotes or slant asymptotes). Discontinuities of Rational Functions Points where the denominator is zero cause vertical asymptotes or holes (removable discontinuities). Graphs of Rational Functions Let's visualize an example rational function: f(x) = (x² - 1) / (x - 2). Show Graph Modeling with Rational Functions Rational functions are used to model rates, ratios, and problems with asymptotic behavior (ex: speed/time). ...

Unit 2: Trigonometry Special Trigonometric Values in the First Quadrant In the first quadrant (0° to 90°), sine, cosine, and tangent are all positive. Key angles: 30°, 45°, 60°. Show Unit Circle Trigonometric Identities on the Unit Circle The unit circle shows that sin²θ + cos²θ = 1 for all angles θ. Inverse Trigonometric Functions Inverse functions (like sin⁻¹x) return angles based on given trigonometric ratios. Law of Sines & Law of Cosines Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) Law of Cosines: c² = a² + b² - 2ab·cos(C) Solving General Triangles Use Law of Sines or Law of Cosines to solve any triangle (not just right triangles). Sinusoidal Equations & Models ...

Unit 6: Vectors Vectors Introduction A vector has both magnitude and direction, represented by an arrow in space. Vector Components A vector can be broken down into horizontal (x) and vertical (y) components. Magnitude of Vectors Magnitude = √(x² + y²) Scalar Multiplication Scaling a vector changes its magnitude but not its direction. Vector Addition and Subtraction Vectors are added by adding their components. Show Vector Addition Direction of Vectors Direction is the angle the vector makes with the x-axis: θ = arctan(y/x) Vector Components from Magnitude and Direction x = magnitude * cos(θ), y = magnitude * sin(θ) ...

Unit 1: Right triangles & trigonometry Unit mastery: 89% Learning Progress: 0% Select a topic: Topic Details Click a topic to learn about it and see visualizations. Adjust angle A (degrees): Score: 0 / 0

Unit 3: Non-right triangles & trigonometry Learning Progress: 0% Select a topic: Topic Details Click a topic to learn about it and see visualizations. Adjust side a: side b: angle C (degrees): Score: 0 / 0

Unit 2: Trigonometric functions Unit mastery: 68% Learning Progress: 0% Select a topic: Topic Details Click a topic to learn about it and see visualizations. Adjust angle (degrees): Score: 0 / 0

Unit 4: Trigonometric equations and identities Unit 4: Trigonometric equations and identities Learning Progress: 0% Select a topic: Topic Details Click a topic to learn about it and see visualizations. Adjust amplitude: frequency: Score: 0 / 0