Function Concepts and Notation • Explore • Lenny.com

Identifying Key Attributes of Quadratic Function Graphs

This video serves as a comprehensive introduction to quadratic functions, specifically focusing on the graphical representation known as a parabola. It bridges the gap between previously learned algebraic concepts—such as intercepts, domain, range, and intervals—and introduces specific attributes unique to quadratics, including the vertex, maximum/minimum values, and the axis of symmetry. The narrator guides viewers through definitions and visual examples, making abstract concepts concrete and accessible for Algebra students. Key themes include mathematical vocabulary acquisition, graphical analysis, and logical reasoning. The video emphasizes the importance of understanding how a function's behavior (increasing or decreasing intervals) determines the shape and orientation of its graph. It also highlights the symmetry inherent in parabolas and how this property can be used to determine equations and missing values even when complete information isn't provided. This resource is highly valuable for Algebra classrooms as it functions not just as a lecture, but as an interactive guided lesson. It explicitly prompts students to pause, complete guided notes, and attempt practice problems before revealing the solutions. This structure supports active learning and self-assessment, making it an excellent tool for introducing the unit, facilitating a flipped classroom model, or providing remediation for students struggling with graphing quadratics.

Miacademy & MiaPrep Learning Channel

13mins 5s

Mastering Linear Equations: Forms, Slopes, and Graphing

This comprehensive video provides a thorough review of linear equations, serving as an excellent resource for students preparing for Algebra exams. It systematically breaks down the three primary forms of linear equations: slope-intercept form, standard form, and point-slope form. The video explains the components of each form, such as coefficients, variables, slope, and intercepts, providing a solid theoretical foundation before moving into practical applications.

The Organic Chemistry Tutor

32mins 5s

How Domain Affects Sequence Formulas

This advanced algebra video explores the relationship between mathematical sequences and their domains, challenging the standard convention that sequences must always begin with the first term at n=1. The narrator demonstrates that sequences can be defined using different starting points for the domain (specifically n=0 vs n=1) as long as the formula is adjusted accordingly. Through clear examples of both arithmetic and geometric sequences, viewers learn how to manipulate explicit and recursive formulas to match specific domain constraints. The video breaks down three specific examples: a doubling geometric sequence starting at 4, a decreasing arithmetic sequence starting at 52, and an alternating geometric sequence starting at 1. For each, the narrator compares valid and invalid formulas, showing how changing the starting value of 'n' changes the structure of the equation. A key segment involves an "imposter" activity where students must analyze four different formulas to identify the one that does not produce the correct sequence based on its defined domain. This resource is highly valuable for high school algebra classrooms as it moves students beyond rote memorization of formulas into a deeper conceptual understanding of functions and domains. It addresses the common student struggle of reconciling different notations for the same pattern. Teachers can use this video to introduce zero-indexing (common in computer science) or to reinforce the importance of checking work by substituting values. It promotes critical thinking by asking students to verify formulas rather than just generate them.

Miacademy & MiaPrep Learning Channel

6mins 58s

Graphing and Transforming Sine and Cosine Functions

This comprehensive mathematics video provides a detailed guide to transforming sine and cosine functions. It systematically breaks down the standard equation y = a*sin(b(x-c)) + d, explaining how each variable affects the graph's shape and position. The video covers key concepts including amplitude, period, phase shifts, and vertical shifts, using clear visual comparisons between parent functions and their transformed counterparts to illustrate these abstract mathematical concepts. The content explores four main types of transformations: vertical stretches/compressions (amplitude), horizontal stretches/compressions (period), horizontal shifts (phase shifts), and vertical shifts (midline adjustments). It also clarifies the relationship between sine and cosine functions, demonstrating how they are essentially the same wave shifted by pi/2. Through step-by-step examples, the video teaches viewers both how to graph a function from a given equation and how to derive an equation from a given graph. For educators, this video is an excellent resource for Algebra II, Trigonometry, or Pre-Calculus units. It transitions from conceptual understanding to procedural fluency, offering pause points for students to predict graph characteristics before solutions are revealed. The inclusion of complex examples that combine multiple transformations makes it suitable for direct instruction, review, or a flipped classroom model, helping students bridge the gap between algebraic formulas and graphical representations.

Miacademy & MiaPrep Learning Channel

21mins 51s

How to Graph Piecewise Functions Step-by-Step

This instructional video provides a comprehensive guide to graphing piecewise functions, covering both fundamental concepts and practical graphing strategies. The lesson begins by defining a piecewise function as a function defined by at least two equations, each applying to a different part of the domain. It distinguishes between the "eraser method"—graphing the entire function and removing unwanted parts—and a more efficient table-based method that focuses strictly on the restricted domain values.

Miacademy & MiaPrep Learning Channel

12mins 12s

How to Calculate Terms in Recursive Sequences

This concise mathematics tutorial introduces students to recursive formulas, a fundamental concept in Algebra and Pre-Calculus regarding sequences. The video demonstrates how to generate the terms of a sequence when given a starting value and a recursive rule. The instructor uses a digital blackboard format to walk through the step-by-step arithmetic required to find subsequent terms based on previous ones, emphasizing the iterative nature of the process. The video explores two key examples: a linear recursive formula involving simple multiplication and addition, and a more complex non-linear formula involving squaring. A significant portion of the video is dedicated to explaining subscript notation (such as n, n+1, and n-1), clarifying how these symbols represent the "current," "next," and "previous" terms relative to one another. This distinction is crucial for students learning to interpret mathematical language. For educators, this resource serves as an excellent direct instruction module or review tool. It isolates the mechanical skill of evaluating recursive functions without the distraction of word problems or complex contexts, making it ideal for introducing the concept or remediating students who struggle with function notation. The clear visual layout helps students track the substitution process, reinforcing algebraic precision.

The Organic Chemistry Tutor

2mins 54s

Understanding One-Sided Limits with Graphs and Piecewise Functions

This educational video provides a comprehensive introduction to one-sided limits, a foundational concept in pre-calculus and calculus. Using a relatable analogy of filling a water balloon until it bursts, the video explains why analyzing a function from only one direction is often necessary. It transitions from this conceptual framework into the formal mathematical notation for left-hand and right-hand limits, explicitly clarifying the difference between the sign of the limit direction and the sign of the number itself.

Miacademy & MiaPrep Learning Channel

16mins 9s

How to Determine Polynomial Signs on Intervals

This instructional video provides a comprehensive guide on how to determine the sign (positive or negative) of a polynomial function within a specific interval. The lesson begins by using graphical representations to build intuition, showing visually how functions dip below or rise above the x-axis between their roots. The instructor, Randy, demonstrates how to identify the relevant section of a graph based on given x-values and read the corresponding y-values to determine the sign.

Miacademy & MiaPrep Learning Channel

10mins

How to Identify Increasing and Decreasing Intervals on Graphs

This educational video provides a clear and structured guide to understanding increasing and decreasing intervals on function graphs. The narrator begins by connecting these concepts to the familiar idea of slope, explaining that an increasing interval corresponds to a positive slope and a decreasing interval to a negative slope. The video also introduces "interval notation," a critical mathematical syntax used to describe specific sections of a domain, explaining the difference between parentheses (exclusive) and square brackets (inclusive), as well as the use of infinity.

Miacademy & MiaPrep Learning Channel

6mins 58s

Identifying Even and Odd Functions: Algebraic and Graphical Tests

This educational video provides a clear and structured explanation of even and odd functions, a fundamental concept in advanced algebra and pre-calculus. The presenter, Randy, guides viewers through the distinct definitions of both function types, distinguishing them from the simpler concepts of even and odd numbers. The video breaks down how to identify these functions using two primary methods: algebraically (by substituting negative inputs into the function) and graphically (by analyzing symmetry). The clear visual aids and step-by-step narration make complex algebraic proofs accessible to students. Key themes include algebraic manipulation, function notation, and graphical symmetry. The video explicitly demonstrates the algebraic tests $f(-x) = f(x)$ for even functions and $f(-x) = -f(x)$ for odd functions. It visualizes these concepts by showing how even functions have symmetry across the y-axis, while odd functions exhibit symmetry about the origin (or rotational symmetry). The lesson concludes with worked examples involving polynomial functions, reinforcing how exponents influence the classification of a function. For the classroom, this video is an excellent resource for introducing or reviewing function properties in Algebra II or Pre-Calculus courses. It addresses the common student misconception that function types are directly analogous to even and odd integers. Teachers can use the step-by-step examples to model how to perform algebraic tests for symmetry, while the graphical demonstrations help build visual intuition. The content supports learning standards related to analyzing functions and understanding key features of graphs.

Miacademy & MiaPrep Learning Channel

9mins 45s

Using Recursive and Explicit Formulas for Arithmetic Sequences

This educational video provides a clear and structured guide to understanding and using formulas for arithmetic sequences. Hosted by a narrator named Justin, the lesson builds upon previous knowledge of what arithmetic sequences are, moving into the practical application of mathematical notation and formulas. The video breaks down the specific symbols used in sequence notation, such as subscripts for term numbers and variables for common differences, before introducing two primary methods for defining sequences: recursive and explicit forms. The content offers a side-by-side comparison of recursive formulas, which rely on knowing the previous term, and explicit formulas, which allow for calculating any term directly. Through step-by-step examples, the narrator demonstrates how to calculate specific terms using both methods. A key highlight of the video is a practical demonstration showing why explicit formulas are superior for finding distant terms (like the 53rd term) compared to the tedious process required by recursive formulas. For educators, this video serves as an excellent instructional tool for Algebra units covering patterns, functions, and sequences. It directly addresses the common student struggle of understanding function-like notation in sequences. The clear distinction between 'finding the next term' and 'finding the nth term' helps students understand efficiency in mathematics. It is suitable for introducing the concepts or for remediation for students struggling to distinguish between the two formula types.

Miacademy & MiaPrep Learning Channel

6mins 28s

How to Evaluate Function Expressions Using Graphs

This educational video serves as a clear, step-by-step tutorial on how to evaluate mathematical expressions involving function notation using graphs. Hosted by a narrator named Justin, the video builds upon prior knowledge of reading function graphs to tackle more complex problems where students must substitute values derived from graphs into arithmetic expressions. It transitions from simple discrete graphs to systems involving multiple continuous functions on the same coordinate plane, demonstrating how to distinguish between different function names like f(x) and g(x). The key themes explored include function notation, interpreting graphical data, substitution, and order of operations. The video specifically emphasizes the process of "finding and replacing" function outputs with their numerical values before simplifying the expression. It addresses potential student anxiety regarding complex-looking equations by breaking them down into manageable steps of identifying inputs and outputs on a grid. For educators, this video is an excellent resource for Algebra I or II classes. It visually connects abstract algebraic notation with concrete graphical representations, helping students visualize what f(x) actually means. The clear color-coding and distinct examples make it useful for introducing the concept of evaluating composite expressions or for reviewing function literacy before moving on to word problems or calculus concepts.

Miacademy & MiaPrep Learning Channel

3mins 30s

Lessons

Algebra Bridge Part 2

Carlos Arenas

Algebra Bridge Part 1

Carlos Arenas

Foundation Missions

Elsa LaLone

The Career Summit

Elsa LaLone

Feature Finder Vocab

Cheydon Krysl

Algebra Bridge Part 2

Cheyenne Watkins

Algebra Bridge Part 1

Cheyenne Watkins

Quarter 4 Algebra Assessment

david.marques

Launch Pad Physics

Aysenur Ersoy

Slope Shifters

deeohnny.jackson

Geometry Regents Mastery

Kayla Borozny

Algebra 1 Regents Mastery

Kayla Borozny

Materials

Launch Pad Focus Companion

Carlos Arenas

Part Two Answer Key

Carlos Arenas

Escape Challenge Workbook

Carlos Arenas

Algebra Preview Workbook

Carlos Arenas

Statistics Probability Workbook

Carlos Arenas

Geometry Foundations Workbook

Carlos Arenas

Linear Scout Workbook

Carlos Arenas

Pattern Pilot Workbook

Carlos Arenas

Launch Pad Answer Key

Carlos Arenas

Proportional Patrol Workbook

Carlos Arenas

Equation Sentinel Workbook

Carlos Arenas

Expression Force Workbook

Carlos Arenas

Sequences

Algebra 1 Launch Pad

Carlos Arenas

Math in Motion: The 2-Hour Summit

Elsa LaLone

Algebra 1 Launch Pad

Cheyenne Watkins

Regents Mastery Suite

Kayla Borozny

Market Math Mastery

david.marques

Market Math Mastery

david.marques

Algebra Essentials Sequence

deeohnny.jackson

MCAS Mastery Geometry

Elsa LaLone

MCAS Mastery Geometry

Elsa LaLone

Log Decoder Sequence

Cheydon Krysl

Math in Motion: The 2-Hour Summit

Elsa LaLone

Proportionality Intervention Track

Brandon Ruffin

Videos

Identifying Key Attributes of Quadratic Function Graphs

Mastering Linear Equations: Forms, Slopes, and Graphing

How Domain Affects Sequence Formulas

Graphing and Transforming Sine and Cosine Functions