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.