Illustrative Math Alignment: Grade 8 Unit 3

Topic

Quadratics

Description

In this example, a quadratic function is shown in tabular and graphic format. Analyzing these different representations of a function is important and provides relevant information about the quadratic function. It demonstrates the role of the quadratic terms in determining the graph's curvature and the significance of intercepts in solving quadratic equations.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Description

In this example, a quadratic function is shown in tabular and graphic format. Analyzing these different representations of a function is important and provides relevant information about the quadratic function. This example shows a downwardly opening parabola that has two roots, one of them shown in the table.

For a complete collection of math examples related to Quadratics click on this link: Math Examples: Quadratics Collection.

Topic

Quadratics

Topic

Quadratics

Description

This example provides a quadratic function displayed in both tabular and graphical forms. The table lists values that, when plotted, create a parabolic curve on the graph. This visual representation helps in identifying key features such as the vertex, axis of symmetry, and roots. By analyzing these features, students can understand the impact of different coefficients on the shape and position of the parabola. Skills involved include interpreting data from tables, graphing functions, and understanding the vertex and standard forms of quadratic equations.

Topic

Quadratics

Description

In this example, a quadratic function is presented through both tabular and graphical forms. The accompanying image provides a visual interpretation of the function, emphasizing the relationship between the x-values and their corresponding y-values. This representation allows learners to recognize the nature of quadratic functions, particularly how changes to the function parameters affect the shape and position of the graph. The skills required include critical graph analysis, understanding symmetry in parabolas, and solving for x-intercepts.

Topic

Quadratics

Description

A detailed view of a quadratic function is depicted here, where the data table and graph work in conjunction to showcase the solution process for the function. Analyzing this illustration aids in comprehending how the quadratic function applies to specific scenarios and enhances one’s ability to derive equations from graphs. Such comparative analysis of multiple representations bolsters problem-solving skills and equips students with the tools to tackle different forms of quadratic equations.

Topic

Quadratics

Description

This image presents a quadratic function in tabular and graphic format. The graph reflects the nature of parabolic functions, while the table reinforces the numerical data needed for graphing. By exploring both representations, students can appreciate the convergence of algebra and geometry, honing their capacity for critical thinking and analysis. Understanding the properties of the parabola is essential for mastery in quadratic equations by applying skills such as completing the square or using the quadratic formula.

Topic

Quadratics

Description

This example consolidates the properties of quadratic functions by using a table to present values and a graph to illustrate their relationship. The significance of identifying intercepts, the vertex, and axis of symmetry becomes apparent when viewed together. Mastering these concepts through multiple formats allows for invaluable preparation for advanced algebraic topics, thereby enhancing overall mathematical comprehension and problem-solving abilities.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / x. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with branches in the first and third quadrants, illustrating the characteristic shape of this basic rational function.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = -1 / (x - 1). The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph includes labeled points at certain coordinates, illustrating how the negative sign in the numerator and the constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x - 1). The image showcases both the table of x-y coordinates and the resulting graph. The graph is plotted with labeled points at given coordinates, illustrating how the negative sign and constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x - 1). The image presents both the table of x-y coordinates and the resulting graph. The graph includes points labeled at specified coordinates, demonstrating how the negative signs in both the numerator and denominator, along with the constant, affect the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (x + 1) + 1. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The table contains specific x-y coordinates: (-2, 0), (-1.5, -1), (0, 2), (1.5, 1.4), (2, 1.333), which are plotted on the graph. This example illustrates how adding a constant to a rational function affects its graph.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (x + 1) - 1. The image showcases both the table of x-y coordinates and the resulting graph. The table contains specific x-y coordinates: (-2, -2), (-1.5, -3), (0, 0), (1.5, -0.6), (2, -0.667), which are plotted on the graph. This example illustrates how subtracting a constant from a rational function affects its graph.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (x + 1) + 1. The image presents both the table of x-y coordinates and the resulting graph. The table contains specific x-y coordinates: (-2, 2), (-1.5, 3), (0, 0), (1.5, 0.6), (2, 0.667), which are plotted on the graph. This example demonstrates how negating the numerator of a rational function and adding a constant affects its graph.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = -1 / (x + 1) - 1. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The table contains specific x-y coordinates: (-2, 0), (-1.5, 1), (0, -2), (1.5, -1.4), (2, -1.333), which are plotted on the graph. This example illustrates how negating the numerator of a rational function and subtracting a constant affects its graph.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x + 1) + 1. The image showcases both the table of x-y coordinates and the resulting graph. Points plotted include (-2, 1.333), (-1.5, 1.4), (0, 2), (1.5, -1), and (2, 0). The graph features a hyperbola with asymptotes, illustrating how negating the x-term in the denominator and adding a constant affects the graph's shape and position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x + 1) - 1. The image presents both the table of x-y coordinates and the resulting graph. Points plotted include (-2, -0.667), (-1.5, -0.6), (0, 0), (1.5, -3), and (2, -2). The graph shows a hyperbola with vertical and horizontal shifts, demonstrating how negating the x-term in the denominator and subtracting a constant affects the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (x - 1) + 1. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. Points include (-2, 0.667), (-1.5, 0.6), (0, 0), (1.5, 3), and (2, 2). The graph is a hyperbola with asymptotes at x = 1 and y = 1, illustrating how subtracting a constant in the denominator and adding a constant to the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / x. The image showcases both the table of x-y coordinates and the resulting graph. The graph is a hyperbola with branches in the second and fourth quadrants, illustrating how the negative sign affects the function's shape compared to the previous example.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (x - 1) - 1. The image showcases both the table of x-y coordinates and the resulting graph. The table contains specific x-y coordinates: (-2, -1.333), (-1.5, -1.4), (0, -2), (1.5, 1), and (2, 0). The graph is a hyperbola with shifted asymptotes at x = 1 and y = -1, illustrating how subtracting a constant in the denominator and from the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = (-1) / (-x + 1) + 1. The image presents both the table of x-y coordinates and the resulting graph. It includes a table of x-y coordinates: (-2, 0.667), (-1.5, 0.6), (0, 0), (1.5, 3), and (2, 2). The graph is plotted on a grid, demonstrating how negating both the numerator and x-term in the denominator, while adding a constant to the entire fraction, affects the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = (-1) / (-x + 1) - 1. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. It includes a table of x-y coordinates: (-2, -1.333), (-1.5, -1.4), (0, -2), (1.5, 1), and (2, 0). The graph is plotted on a grid, illustrating how negating both the numerator and x-term in the denominator, while subtracting a constant from the entire fraction, affects the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = (-1) / (x - 1) + 1. The image shows both a table of x-y coordinates and the corresponding graph. The table includes points such as (-2, 1.333), (-1.5, 1.4), (0, 2), (1.5, -1), and (2, 0). The graph is plotted on a grid, illustrating how subtracting a constant in the denominator and adding a constant to the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = (-1) / (x - 1) - 1. The image shows both a table of x-y coordinates and the corresponding graph. The table includes points such as (-2, -0.667), (-1.5, -0.6), (0, 0), (1.5, -3), and (2, -2). The graph is plotted on a grid, illustrating how subtracting constants both in the denominator and from the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x - 1) + 1. The image showcases both the table of x-y coordinates and the resulting graph. The graph shows points plotted at (-2, 2), (-1.5, 3), (0, 0), (1.5, 0.6), and (2, 0.667), illustrating how negating both terms inside the parentheses and adding a constant to the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (-x - 1) - 1. The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph shows points plotted at (-2, 0), (-1.5, 1), (0, -2), (1.5, -1.4), and (2, -1.333), illustrating how negating both terms inside the parentheses and subtracting a constant from the entire fraction affects the graph's shape and position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x - 1) + 1. The image presents both the table of x-y coordinates and the resulting graph. The graph shows points plotted at (-2, 0), (-1.5, -1), (0, 2), (1.5, 1.4), and (2, 1.333), demonstrating how negating both the numerator and the terms inside the parentheses, while adding a constant to the entire fraction, affects the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x - 1) - 1. The image showcases both the table of x-y coordinates and the resulting graph. The graph shows points plotted at (-2, -2), (-1.5, -3), (0, 0), (1.5, -0.6), and (2, -0.667), illustrating how negating both the numerator and the terms inside the parentheses, while subtracting a constant from the entire fraction, affects the graph's shape and position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (-x). The image presents both the table of x-y coordinates and the resulting graph. Interestingly, the graph is identical to that of y = -1 / x, showing a hyperbola with branches in the second and fourth quadrants. This example demonstrates how different forms of rational functions can produce the same graph.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x). The image showcases both the table of x-y coordinates and the resulting graph. Notably, the graph is identical to that of y = 1 / x, displaying a hyperbola with branches in the first and third quadrants. This example further illustrates how different forms of rational functions can produce the same graph.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (x + 1). The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with points marked at specific coordinates, illustrating how the addition of a constant in the denominator affects the graph's position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (x + 1). The image showcases both the table of x-y coordinates and the resulting graph. The graph displays a hyperbola with labeled points on the curve, illustrating how the negative sign in the numerator and the constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example focuses on creating a table of x-y coordinates and graphing the function y = 1 / (-x + 1). The image provided shows both a table of x-y coordinates and the corresponding graph of the function. The graph is a hyperbola with specific points highlighted, illustrating how the negative sign and constant in the denominator affect the graph's shape and position.

Topic

Rational Functions

Description

This math example demonstrates the creation of a table of x-y coordinates and the graphing of the function y = 1 / (x - 1). The image showcases both the table of x-y coordinates and the resulting graph. The graph features a hyperbola with marked points on its curve, illustrating how the constant in the denominator affects the graph's position.

Topic

Rational Functions

Description

This math example illustrates the creation of a table of x-y coordinates and the graphing of the function y = -1 / (-x + 1). The image presents both the table of x-y coordinates and the resulting graph. The graph is plotted with points labeled at specific coordinates, demonstrating how the negative signs in both the numerator and denominator affect the graph's shape and position.

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions

Topic

Special Functions