Graph Equations and Inequalities

This instructional video guides students through the process of graphing linear equations provided in standard form (Ax + By = C). Hosted by a narrator named Justin, the lesson explores two primary strategies: finding the x and y-intercepts and rearranging the equation into slope-intercept form. The video emphasizes procedural fluency while also encouraging critical thinking about efficiency and choosing the best method for specific problems.

This comprehensive math tutorial demystifies piecewise functions by connecting abstract algebra concepts to real-world scenarios. The video begins with a relatable "Buy More, Save More" retail example to intuitively explain why a single function might require different rules for different input values. It progresses into formal mathematical definitions, demonstrating how to evaluate function values based on specific domain intervals and how to handle inequalities. The core of the lesson provides a step-by-step walkthrough of graphing a complex piecewise function that includes linear, absolute value, quadratic, and constant components. It pays special attention to common student stumbling blocks, such as distinguishing between open and closed circles at boundary points and handling gaps in the domain. The video also teaches students how to determine the overall range of a piecewise function by analyzing the output intervals of its individual parts. Finally, the video scaffolds the advanced skill of "reverse engineering" a function—writing the algebraic equations when given a graph. By reinforcing prior knowledge of slope-intercept form and function transformations, this resource serves as an excellent primary lesson for Algebra 2 students or a robust review for Pre-Calculus students encountering these "Frankenstein" functions.

This comprehensive mathematics tutorial guides students through the process of graphing rational functions, bridging the gap between algebraic analysis and visual representation. Starting with the parent reciprocal function 1/x, the video systematically explores how to apply transformations, identify domain restrictions, and determine the behavior of graphs near asymptotes. It moves from simple vertical and horizontal shifts to complex functions requiring factoring to find holes (removable discontinuities), vertical asymptotes, and oblique asymptotes. The video covers key themes essential for Algebra II and Pre-Calculus students, including reciprocal functions, coordinate plane analysis, and the behavior of functions at undefined points. It introduces specific strategies for graphing without technology, such as finding intercepts using constant terms and selecting strategic test points near asymptotes to determine curve direction. The distinction between 1/x and 1/x² is analyzed to explain why certain graphs occupy specific quadrants. For educators, this resource serves as an excellent instructional core or review tool. It breaks down a complex multi-step procedure into manageable chunks, providing pause points for independent practice. The video encourages critical thinking by asking students to predict graph shapes based on algebraic features before plotting points, fostering a deeper conceptual understanding of the link between an equation's structure and its graphical form.

This educational video provides a clear, step-by-step guide to understanding linear graphs, slope, and the critical distinction between dependent and independent variables. Narrated by Kaylee, the lesson begins by reviewing how to calculate slope using "rise over run" on a coordinate grid, demonstrating how a constant slope allows points to form a straight line. It then transitions into real-world applications, using examples like running speed and dog walking wages to show how linear relationships appear in everyday life.

This comprehensive tutorial video walks students through ten of the most fundamental and frequently encountered problem types in a standard high school or college Algebra course. The instructor methodically demonstrates how to solve specific examples ranging from calculating slope and solving linear equations to more complex tasks like factoring trinomials, graphing parabolas, and solving systems of equations. Each problem is presented clearly on a blackboard-style interface, with the instructor breaking down the logic and steps required to reach the solution. The video explores key algebraic themes including linear functions, inequalities, quadratic equations, exponents, and systems of equations. It serves as an excellent review of the core curriculum found in Algebra I and Algebra II. The instructor emphasizes not just the 'how' but often the 'why,' discussing concepts like the axis of symmetry, properties of exponents, and different methods for solving quadratics (such as factoring versus the quadratic formula). For educators, this video is a versatile tool that can be used for end-of-unit reviews, standardized test preparation (like the SAT or ACT), or flipped classroom assignments. Teachers can assign specific segments to reinforce struggling concepts or use the entire video as a comprehensive course review. The clear, step-by-step visual working makes it ideal for students who benefit from seeing mathematical processes modeled explicitly.

This video provides a detailed, step-by-step tutorial on how to graph reciprocal functions using the standard equation form y = a/(x-h) + k. The instructor explains the role of each variable in the equation, specifically focusing on how 'h' and 'k' determine the vertical and horizontal asymptotes, and how 'a' influences the vertical stretch, shrink, and orientation of the graph across the coordinate plane. The video explores key themes such as identifying asymptotes, understanding transformations (shifts and stretches), and the process of creating a table of values to plot accurate points. It contrasts positive and negative values of 'a', demonstrating how a negative sign reflects the graph across the horizontal asymptote, changing which quadrants the curves occupy relative to the asymptotes. For educators, this video serves as an excellent instructional tool for Algebra II or Pre-Calculus students moving beyond linear and quadratic functions. It offers a clear procedural method for graphing rational functions without a calculator, reinforcing concepts of domain, range, and asymptotic behavior. The step-by-step examples allow teachers to pause and let students practice predicting the graph's behavior before the solution is revealed.

This educational math video provides a clear, step-by-step guide to understanding function reflections both graphically and algebraically. The narrator, Randy, uses visual aids and specific examples to demonstrate how functions behave when reflected across the x-axis, the y-axis, and the origin (both axes simultaneously). The video breaks down complex notation into digestible concepts, showing exactly how changes in function notation correspond to visual shifts on a coordinate plane. The content focuses on three primary transformation types: negative f(x), f of negative x, and negative f of negative x. Using the square root function as a base visual model, the video illustrates how these changes move a graph from one quadrant to another. It then transitions into more complex algebraic applications involving polynomial functions, teaching students how to distribute negatives and handle exponents when transforming equations. For educators, this video serves as an excellent instructional tool for Algebra II or Pre-Calculus units on function transformations. It bridges the gap between abstract algebraic manipulation and concrete visual understanding. Teachers can use the included examples—one purely algebraic involving a cubic polynomial and one purely graphical involving parabolas—to check for student understanding and scaffold learning from simple to complex tasks.

This video provides a clear, step-by-step visual guide on how to graph a linear function given in slope-intercept form. The narrator demonstrates the process using the specific example function f(x) = -3/4x + 2, beginning by demystifying function notation and rewriting the equation with 'y'. The tutorial methodically breaks down the equation into its key components: the y-intercept (b) and the slope (m), explaining how each determines the position and direction of the line on a coordinate plane. Key themes include understanding the relationship between f(x) and y, identifying parts of the slope-intercept form (y = mx + b), and applying the concept of 'rise over run' to plot points. The video specifically addresses how to handle negative slopes and fractional slopes, showing how to move 'down and to the right' to create a descending line. It also demonstrates how to extend the line in the opposite direction to ensure accuracy. For educators, this video serves as an excellent instructional tool for introducing or reviewing graphing linear equations in Algebra I or Pre-Algebra courses. Its visual approach—using animated arrows to show the 'rise' and 'run' movements—helps students conceptualize slope as a rate of change rather than just a number. It effectively addresses common stumbling blocks, such as interpreting function notation and graphing negative fractional slopes, making it valuable for visual learners.

This instructional video provides a clear, step-by-step tutorial on how to determine the vertex and axis of symmetry for absolute value functions. The narrator demonstrates both algebraic and graphical methods, starting with basic examples and progressing to more complex equations. Viewers learn the fundamental technique of setting the expression inside the absolute value bars to zero to find the axis of symmetry, and then substituting that value back into the equation to find the vertex's y-coordinate. The video explores key themes in Algebra and pre-calculus, specifically focusing on the properties of absolute value graphs. It covers the general shape of these functions (the "V" shape), how to determine if the graph opens upward or downward based on the sign of the leading coefficient, and the relationship between the algebraic equation and its visual representation on a coordinate plane. The tutorial concludes by generalizing these concepts into the standard vertex form equation. For educators, this video serves as an excellent resource for scaffolding lessons on functions and transformations. It is particularly useful for visual learners who benefit from seeing the connection between algebraic manipulation and graphical outcomes. Teachers can use this video to introduce the topic, reinforce homework concepts, or help students self-remediate when struggling with function transformations. The clear handwriting and paced explanation make it easy to pause and practice alongside the narrator.

This comprehensive mathematics tutorial provides a clear, step-by-step guide on how to classify mathematical functions as even, odd, or neither. The video breaks down the concept into two primary methods: algebraic analysis and graphical interpretation. It begins by defining the formal algebraic conditions for even functions ($f(-x) = f(x)$) and odd functions ($f(-x) = -f(x)$), while also introducing a practical "shortcut" involving the exponents of the variables. The narrator walks through multiple algebraic examples, demonstrating how to substitute negative variables and factor equations to prove a function's classification. The second half of the video shifts to visual learning, explaining how symmetry on a coordinate plane identifies function types. It demonstrates that even functions are symmetric about the y-axis, while odd functions are symmetric about the origin. The video illustrates these concepts with hand-drawn graphs of parabolas, cubic functions, lines, and complex polynomials. It visually emphasizes how translations (shifts) affect symmetry and changes a function's classification from even/odd to "neither." For educators and students, this resource is invaluable for Algebra II and Pre-Calculus curricula. It addresses common stumbling blocks, such as how to handle constants (treating them as $x^0$) and linear terms (treating $x$ as $x^1$). The video concludes with a critical conceptual test using a circle, challenging students to recall the definition of a function itself (the Vertical Line Test) before attempting to classify it. This encourages critical thinking beyond rote memorization of rules.

This detailed mathematics tutorial explains the fundamental concepts of graphing circles from algebraic equations. It begins by introducing the standard equation of a circle centered at the origin and expands to circles with shifted centers using the formula (x-h)² + (y-k)² = r². The video demonstrates how to extract the center coordinates and radius from an equation to accurately plot the shape on a coordinate plane.

This comprehensive math tutorial serves as a rigorous review of the Polynomial Graphs unit, guiding students through the essential techniques for analyzing and graphing polynomial functions. The video systematically revisits core algebraic skills, including finding zeros of both factored and non-factored polynomials, factoring cubic expressions, and using number lines to determine the sign of functions over specific intervals. The content advances into the graphical characteristics of polynomials, exploring how the multiplicity of zeros determines whether a graph crosses or bounces at the x-axis. It further explains how to determine end behavior based on the leading coefficient and degree, and concludes with practical applications of the Remainder Theorem and Factor Theorem to identify roots and factors. Ideal for Algebra II and Pre-Calculus classrooms, this video connects algebraic manipulation with visual understanding. It effectively models problem-solving strategies for identifying functions from graphs and vice versa, making it a valuable resource for unit reviews, exam preparation, or remediating complex topics regarding polynomial behavior.