Illustrative Math Alignment: Grade 6 Unit 7

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x - 8) / ((x + 3)(x - 3)). The graph features vertical asymptotes at x = -3 and x = 3, highlighted by dashed red lines. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = -(x + 1) / (x + 2). The graph features a single vertical asymptote at x = -2, indicated by a dashed red line. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -1 / x. The graph is a hyperbola with vertical and horizontal asymptotes, with the vertical asymptote at x = 0. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = -(x - 3) / (x + 4). The graph features a vertical asymptote at x = -4, but it is incorrectly labeled as x = -2. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x - 5) / (x - 4). The graph features a vertical asymptote at x = 4, which is correctly labeled. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = -(x + 2) / ((x + 4)(x + 1)). The graph features vertical asymptotes at x = -4 and x = -1. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x - 4) / ((x + 5)(x + 2)). The graph features vertical asymptotes at x = -5 and x = -2. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = -(x + 2) / ((x - 3)(x + 7)). The graph features vertical asymptotes at x = -7 and x = 3. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x + 1) / ((x + 2)(x - 8)). The graph features vertical asymptotes at x = -2 and x = 8. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = -(x - 4) / ((x - 3)(x + 2)). The graph features vertical asymptotes at x = -2 and x = 3. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = -(x - 3) / ((x + 3)(x - 1)). The graph features vertical asymptotes at x = -3 and x = 1. Students are tasked with graphing the function and determining its vertical asymptotes.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = -(x - 1) / ((x - 2)(x - 3)). The graph features vertical asymptotes at x = 2 and x = 3, marked with dashed red lines on the coordinate plane. Students are asked to graph the function and identify its vertical asymptotes.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = 1 / (x + 2). The graph is a hyperbola with a vertical asymptote at x = -2. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = 1 / (x - 4). The graph is a hyperbola with a vertical asymptote at x = 4. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = 1 / (5x + 3). The graph features a vertical asymptote at x = -0.6, indicated by a dashed red line. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example demonstrates the graph of the rational function y = -1 / (6x + 2). The graph features a vertical asymptote at x = -1/3, shown with a dashed red line on the grid. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example showcases the graph of the rational function y = 1 / (5x - 7). The graph features a vertical asymptote at x = 1.4, marked by a dashed red line on the grid. Students are tasked with graphing the function and determining its vertical asymptote.

Topic

Rational Functions

Description

This example presents the graph of the rational function y = 1 / (-9x + 3). The graph is plotted on a grid with a vertical asymptote at x = 1/3, marked by a dashed red line. Students are asked to graph the function and identify its vertical asymptote.

Topic

Rational Functions

Description

This example illustrates the graph of the rational function y = (x + 3) / (x + 2). The graph features a vertical asymptote at x = -2 and shows two branches, one in the second quadrant and one in the fourth quadrant. Students are tasked with graphing the function and determining its vertical asymptote.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

This is part of a collection of math examples that focus on rational number concepts. This includes rational numbers, expressions, and functions.

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.