Illustrative Math Alignment: Grade 8 Unit 3

Solve an interactive word search puzzle on the topic of Linear Equations and Functions.

Related Resources

Solve an interactive word search puzzle on the topic of Linear Equations and Functions.

Related Resources

Lesson Plan: Linear Functions and Applications

This is the fifth and final lesson in a five-part series on linear equations and inequalities designed for middle school students. In this lesson, students explore linear functions in-depth, learning how to represent them using equations, tables, and graphs. They will analyze key function properties, including slope, y-intercept, domain, and range, to develop a complete understanding of how linear functions behave.

The lesson focuses on real-world applications, demonstrating how linear functions are used in various fields such as finance, physics, and business. Students will solve problems involving rates of change, cost analysis, and motion, reinforcing the practical significance of linear functions. Through guided practice and hands-on activities, they will interpret linear relationships and predict outcomes using function models.

In this lesson, students will explore direct variation, a special type of proportional relationship where one variable is a constant multiple of another. They will learn how to identify direct variation in tables, graphs, and equations, determine the constant of variation ($k$), and understand how direct variation relates to linear functions.

Key concepts covered in this lesson include:

In this lesson, students explore the properties of parallel and perpendicular lines in coordinate geometry. They will learn how to identify these lines based on their slopes, determine equations for parallel and perpendicular lines, and apply these concepts to solve real-world and mathematical problems.

Key concepts covered in this lesson include:

This lesson is the third installment in Media4Math's comprehensive six-part series on understanding slope in mathematics. It focuses on helping students visualize and comprehend slope by engaging in hands-on activities that involve plotting points on a coordinate plane and analyzing the resulting lines.

This lesson is the sixth and final installment in Media4Math's comprehensive six-part series on understanding slope in mathematics. It focuses on interpreting slope as a rate of change between two quantities, enabling students to apply this concept to real-world scenarios and mathematical problems.

Topic

Exponential Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the exponential function y = 1 * 2x. The table lists coordinates for x values from -2 to 2, showing how the y values grow exponentially. The corresponding graph visually represents this rapid growth, with points plotted at (-2, 0.25), (-1, 0.5), (0, 1), (1, 2), and (2, 4).

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 1 * 2-0.3x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a fractional exponent.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = -0.4 * 2-0.6x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a negative coefficient and a fractional exponent.

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -0.5 * 2-x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a negative coefficient and a negative exponent.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = -1 * 2-0.5x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a negative coefficient and a fractional negative exponent.

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 0.5 * 2x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a fractional coefficient.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = 1 * 20.7x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a fractional positive exponent.

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 0.4 * 20.9x. The image provides both a table of values and a graph plotted on a Cartesian plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a fractional coefficient and a fractional exponent.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = 0.6 * 2-x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a fractional coefficient and a negative exponent.

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -1 * 20.8x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of a decreasing exponential function with a negative coefficient and a fractional positive exponent.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = 1 * 10x. The image presents both a table of values and a graph plotted on a coordinate plane, with points labeled for x values from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a base of 10.

Topic

Exponential Functions

Description

This example presents the exponential function y = 4 * 2x through a table of x-y coordinates and its corresponding graph. The table shows five key points: (-2, 1), (-1, 2), (0, 4), (1, 8), and (2, 16). These points are then plotted on a graph, demonstrating how the function grows exponentially as x increases.

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 4 * 10x. The image provides both a table of values and a graph plotted on a coordinate plane, with points labeled for x values ranging from -2 to 2. This representation helps students visualize the behavior of an increasing exponential function with a coefficient greater than 1 and a base of 10.

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = 1 * 102x. The image displays both a graph and a corresponding table, illustrating the rapid growth of this function. The table includes key points such as (-2, 0.0001), (-1, 0.01), (0, 1), (1, 100), and (2, 10,000), demonstrating how quickly the y-values increase as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 3 * 103x. The image showcases both a graph and a corresponding table, highlighting the extremely rapid growth of this function. The table includes key points such as (-2, 3 * 10-6), (-1, 0.003), (0, 3), (1, 3000), and (2, 3 * 106), illustrating the dramatic increase in y-values as x increases.

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = -1 * 10-x. The image displays both a graph and a corresponding table, illustrating the behavior of this negative exponential function. The table includes key points such as (-2, -100), (-1, -10), (0, -1), (1, -0.1), and (2, -0.01), demonstrating how the y-values approach zero as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -3 * 10-x. The image showcases both a graph and a corresponding table, highlighting the behavior of this negative exponential function with a coefficient. The table includes key points such as (-2, -300), (-1, -30), (0, -3), (1, -0.3), and (2, -0.03), illustrating how the y-values approach zero from the negative side as x increases.

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = -1 * 10-2x. The image displays both a graph and a corresponding table, illustrating the behavior of this negative exponential function with a doubled exponent. The table includes key points such as (-2, -10000), (-1, -100), (0, -1), (1, -0.01), and (2, -1 * 10-4), demonstrating how rapidly the y-values approach zero from the negative side as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -3 * 10-2x. The image showcases both a graph and a corresponding table, highlighting the behavior of this negative exponential function with a coefficient and a doubled exponent. The table includes key points such as (-2, -30000), (-1, -300), (0, -3), (1, -0.03), and (2, -3 * 10-4), illustrating how rapidly the y-values approach zero from the negative side as x increases.

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = -0.4 * 10x. The image displays both a graph and a corresponding table, illustrating the behavior of this negative exponential function with a fractional coefficient. The table includes key points such as (-2, -0.004), (-1, -0.04), (0, -0.4), (1, -4), and (2, -40), demonstrating how the y-values decrease rapidly in the negative direction as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = 1 * 100.5x. The image showcases both a graph and a corresponding table, highlighting the behavior of this exponential function with a fractional exponent. The table includes key points such as (-2, 0.1), (-1, 0.316), (0, 1), (1, 3.162), and (2, 10), illustrating how the y-values increase at a slower rate compared to a standard exponential function.

Topic

Exponential Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the exponential function y = -0.5 * 10-0.3x. The image displays both a graph and a corresponding table, illustrating the behavior of this negative exponential function with a fractional coefficient and a negative fractional exponent. The table includes key points such as (-2, -1.99), (-1, -0.998), (0, -0.5), (1, -0.251), and (2, -0.126), demonstrating how the y-values approach zero from the negative side as x increases.

Topic

Exponential Functions

Description

This example showcases the exponential function y = 1 * 23x through a table of x-y coordinates and its corresponding graph. The table presents five key points: (-2, 0.02), (-1, 0.13), (0, 1), (1, 8), and (2, 64). These points are then plotted on a graph, illustrating the rapid growth of this function as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -0.6 * 10-x. The image showcases both a graph and a corresponding table, highlighting the behavior of this negative exponential function with a fractional coefficient and a negative exponent. The table includes key points such as (-2, -60), (-1, -6), (0, -0.6), (1, -0.06), and (2, -0.006), illustrating how rapidly the y-values approach zero from the negative side as x increases.

Topic

Exponential Functions

Description

This math example demonstrates the exponential function y = -1 * 10-0.7x in both tabular and graphical form. The table provides x-y coordinates for x values ranging from -2 to 2, showing how the function's output changes as x increases. The corresponding graph visually represents this relationship, illustrating the characteristic shape of an exponential decay function.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = -0.4 * 10x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating how the function's output changes rapidly as x increases. The accompanying graph provides a visual representation of this exponential growth function, highlighting its distinctive shape and behavior.

Topic

Exponential Functions

Description

This math example showcases the exponential function y = 1 * 100.6x through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, illustrating how the function's output changes as x increases. The graph provides a visual representation of this exponential growth function, emphasizing its characteristic shape and behavior.

Topic

Exponential Functions

Description

This math example presents the exponential function y = 0.6 * 100.8x through both a table of x-y coordinates and a graph. The table provides values for x ranging from -2 to 2, demonstrating how the function's output changes as x increases. The accompanying graph offers a visual representation of this exponential growth function, highlighting its distinctive shape and behavior.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = 0.8 * 10-x through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, showing how the function's output changes as x increases. The graph provides a visual representation of this exponential decay function, emphasizing its characteristic shape and behavior.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = -1 * 10-0.3x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating how the function's output changes as x increases. The accompanying graph provides a visual representation of this exponential decay function with a negative coefficient, highlighting its distinctive shape and behavior.

Topic

Exponential Functions

Description

This math example presents the exponential function y = 1 * ex through both a table of x-y coordinates and a graph. The table provides values for x ranging from -2 to 2, demonstrating how the function's output changes as x increases. The accompanying graph offers a visual representation of this natural exponential function, highlighting its characteristic shape and behavior.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = 5 * ex through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, showing how the function's output changes as x increases. The graph provides a visual representation of this scaled natural exponential function, emphasizing its characteristic shape and behavior.

Topic

Exponential Functions

Description

This math example showcases the exponential function y = 1 * e3x through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, illustrating how the function's output changes rapidly as x increases. The graph provides a visual representation of this exponential growth function with a modified exponent, emphasizing its steep growth curve and behavior.

Topic

Exponential Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the exponential function y = 3 * 22x. The image provides a visual representation of the function, showing both a table of coordinates and a graph plotted on a coordinate plane. The table lists x-y coordinates for x values ranging from -2 to 2, with corresponding y values calculated using the given function.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = 6 * e2x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating the rapid growth of this function as x increases. The accompanying graph provides a visual representation of this exponential growth function, highlighting its steep curve and behavior.

Topic

Exponential Functions

Description

This math example showcases the exponential function y = -1 * e-x through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, illustrating how the function's output changes as x increases. The graph provides a visual representation of this exponential decay function with a negative coefficient, emphasizing its unique shape and behavior.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = -7 * e-x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating how the function's output changes as x increases. The accompanying graph provides a visual representation of this exponential decay function with a negative coefficient, highlighting its distinctive shape and behavior.

Topic

Exponential Functions

Description

This math example showcases the exponential function y = -1 * e-2x through a table of x-y coordinates and a corresponding graph. The table lists values for x ranging from -2 to 2, illustrating how the function's output changes rapidly as x increases. The graph provides a visual representation of this exponential growth function with a negative coefficient, emphasizing its unique shape and behavior.

Topic

Exponential Functions

Description

This math example illustrates the exponential function y = -5 * e-3x through both a table of values and a graph. The table presents x-y coordinates for x values from -2 to 2, demonstrating the extremely rapid change in the function's output as x increases. The accompanying graph provides a visual representation of this exponential growth function with a negative coefficient and tripled exponent, highlighting its steep curve and unique behavior.

Topic

Exponential Functions

Description

This example showcases the exponential function y = -0.5 * ex through a graph and table of x-y coordinates. The table provides points (-2, -0.068), (-1, -0.184), (0, -0.5), (1, -1.36), and (2, -3.69), which are plotted on the accompanying graph. Students are tasked with creating this table and graph to visualize the function's behavior.

Topic

Exponential Functions

Description

This example showcases the exponential function y = 1 * e-0.5x through a graph and table of x-y coordinates. The table provides points (-2, 2.718), (-1, 1.648), (0, 1), (1, 0.607), and (2, 0.368), which are plotted on the accompanying graph. Students are tasked with creating this table and graph to visualize the function's behavior.

Topic

Exponential Functions

Description

This example illustrates the exponential function y = -0.6 * e-0.7x through a graph and table of x-y coordinates. The table provides points (-2, -2.433), (-1, -1.208), (0, -0.6), (1, -0.298), and (2, -0.148), which are plotted on the accompanying graph. Students are challenged to create this table and graph to visualize the function's behavior.

Topic

Exponential Functions

Description

This example presents the exponential function y = -0.5 * e-x through a graph and table of x-y coordinates. The table provides points (-2, -3.695), (-1, -1.359), (0, -0.5), (1, -0.184), and (2, -0.068), which are plotted on the accompanying graph. Students are tasked with creating this table and graph to visualize the function's behavior.