Illustrative Math Alignment: Grade 8 Unit 3

Topic

Linear Functions

Description

This image continues from the previous one, now labeling two coordinates on the graph and showing the general form of the slope formula. It emphasizes that typically, the slope formula is used when two points on the line are known.

By using this visual aid, teachers can help students understand how the slope formula relates to points on a line. This image is particularly useful for reinforcing the concept of slope calculation before introducing the point-slope form.

Topic

Linear Functions

Description

This image presents a variation of the previous graph, now with one coordinate highlighted. It introduces the case where the coordinates of one point are known, along with the slope of the line. This scenario sets the stage for introducing the point-slope form of a linear equation.

By incorporating this visual aid, teachers can help students transition from the two-point slope formula to the point-slope form. This image is particularly useful for demonstrating how different pieces of information about a line can be used to determine its equation.

Topic

Linear Functions

Description

This image continues from the previous one, showing a rearrangement of the slope formula to derive the point-slope form. It demonstrates how algebraic manipulation can lead to a new, useful form of a linear equation.

By using this visual aid, teachers can guide students through the process of deriving the point-slope form. This image is particularly useful for students who benefit from seeing the step-by-step transformation of mathematical expressions.

Topic

Linear Functions

Description

This image continues from the previous one, now showing the single coordinate values plugged into the point-slope equation. It demonstrates how to use the point-slope form when you know the coordinates of one point and the slope of the line.

By incorporating this visual aid, teachers can help students understand how to apply the point-slope form in practice. This image is particularly useful for bridging the gap between the abstract form of the equation and its practical application.

Topic

Linear Functions

Description

This image presents a graph crossing the origin and the corresponding point-slope form. It provides the simplest example of using the point-slope form, where a line passes through the origin (0,0) with a slope of 1.

By using this visual aid, teachers can help students understand the point-slope form in its most basic application. This image is particularly useful for building confidence before moving on to more complex examples.

Topic

Linear Functions

Description

This image presents a variation of the previous example, now with the coordinate (0, 1) and a slope of 1. It demonstrates how to use the point-slope form when the y-intercept is not at the origin.

By incorporating this visual aid, teachers can help students understand how the point-slope form can be used with different y-intercepts. This image is particularly useful for showing how the form adapts to different scenarios while maintaining its basic structure.

Topic

Linear Functions

Description

This image shows a variation of the previous example, now with a line having a negative slope of -2 and crossing the y-axis at (0, -2). It demonstrates how to use the point-slope form with negative slopes and y-intercepts.

By using this visual aid, teachers can help students understand how the point-slope form accommodates negative values. This image is particularly useful for expanding students' understanding of the form's versatility.

Topic

Linear Functions

Description

This math clip art image is part of a series illustrating key linear function concepts. It depicts a bicycle alongside a linear graph, demonstrating the concept of constant speed. The graph shows a linear function that crosses the origin and has a positive slope, representing a scenario where at the starting point (t = 0), no distance has been traveled.

Topic

Linear Functions

Description

This math clip art image is part of a series illustrating key linear function concepts. It features a taxi alongside a graph labeled "Situation: Cost of a Taxi". The graph represents a linear function with a positive y-intercept, demonstrating a scenario with a base fee and a rate per mile traveled.

Topic

Linear Functions

Description

This math clip art image is part of a series illustrating key linear function concepts. It depicts a person walking a dog alongside a graph labeled "Situation: Walking Home". The graph shows a linear function with a negative slope, representing a scenario where the distance from home decreases as time increases.

Topic

Linear Functions

Description

This image is part of a series illustrating key concepts in linear functions, which are fundamental to understanding how lines behave in mathematics. The specific content of this image features a mountain and a graph titled "Situation: Altitude Change". It demonstrates a linear function with a negative slope, represented by the equation y = -0.01x + 101, showing how air pressure changes with altitude.

Topic

Linear Functions

Description

This image is part of a series illustrating key concepts in linear functions, which are essential for understanding how lines behave in mathematics. The specific content of this image features sunflowers and a graph titled "Situation: Plant Growth". It demonstrates a linear function with a positive slope, represented by the equation y = 2x + 20, showing how the height of a sunflower changes over time.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 2x. This involves creating a table of values for x and f(x) using f(x) = 2x. The output for each x is calculated by multiplying x by 2. The results are f(x) = -4, -2, 0, 2, and 4 for x = -2, -1, 0, 1, and 2, respectively.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 3x + 1. This involves creating a table of values for x and f(x) using f(x) = 3x + 1. The output for each x is calculated by tripling x and then adding 1. The results are f(x) = -5, -2, 1, 4, and 7 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = 4 * x + 1. This involves a table created for x and f(x) values. By substituting each x value into f(x) = 4 * x + 1, the outputs are calculated as follows: -2 gives -7, -1 gives -3, 0 gives 1, 1 gives 5, and 2 gives 9. The common difference between successive outputs is +4, confirming a linear function.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = 5 * x + 1. This involves a table created for x and f(x) values. substituting each x into f(x) = 5 * x + 1, the outputs are calculated: -2 gives -9, -1 gives -4, 0 gives 1, 1 gives 6, and 2 gives 11. the common difference between outputs is +5, confirming a linear function.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = -2 * x + 1. This involves a table created for x and f(x) values. Substituting each x into f(x) = -2 * x + 1, the outputs are: -2 gives 5, -1 gives 3, 0 gives 1, 1 gives -1, and 2 gives -3. The common difference is -2, indicating a linear function.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = -3 * x + 1. This involves a table created for x and f(x) values. Each x substituted into f(x) = -3 * x + 1 yields: -2 gives 7, -1 gives 4, 0 gives 1, 1 gives -2, and 2 gives -5. The common difference is -3, confirming linearity.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = -4 * x + 1. This involves a table is created for x and f(x) values. Calculating for each x: -2 gives 9, -1 gives 5, 0 gives 1, 1 gives -3, and 2 gives -7. The common difference is -4, indicating linearity.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the equation f(x) = -5 * x + 1. This involves a table created for x and f(x) values. Substituting into f(x) = -5 * x + 1, the outputs are: -2 gives 11, -1 gives 6, 0 gives 1, 1 gives -4, and 2 gives -9. The common difference is -5, confirming linearity.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the constant function f(x) = 2. This involves a table created for x and f(x) values, where f(x) = 2 for all x values, resulting in outputs of 2, 2, 2, 2, and 2. The common difference is 0, indicating a constant function: y = 2.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the constant function f(x) = 0. This involves a table created for x and f(x) values, where f(x) = 0 for all x values, giving outputs of 0, 0, 0, 0, and 0. The common difference is 0, indicating a constant function: y = 0.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using the function machine represented by the constant function f(x) = -1. This involves a table created for x and f(x) values, where f(x) = -1 for all x values, resulting in outputs of -1, -1, -1, -1, and -1. The common difference is 0, indicating a constant function: y = -1.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 3x. This involves creating a table of values for x and f(x) using f(x) = 3x. The output for each x is calculated by multiplying x by 3. the results are f(x) = -6, -3, 0, 3, and 6 for x = -2, -1, 0, 1, and 2, respectively.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 4x. This involves creating a table of values for x and f(x) using f(x) = 4x. The output for each x is calculated by multiplying x by 4. The results are f(x) = -8, -4, 0, 4, and 8 for x = -2, -1, 0, 1, and 2, respectively.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 5x. This involves creating a table of values for x and f(x) using f(x) = 5x. The output for each x is calculated by multiplying x by 5. The results are f(x) = -10, -5, 0, 5, and 10 for x = -2, -1, 0, 1, and 2, respectively.

Linear Functions are a key concept in mathematics that involves understanding the relationship between input and output values based on a given rule. Examples like this one help students visualize and analyze patterns, making it easier to comprehend linear relationships.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = -2x. This involves creating a table of values for x and f(x) using f(x) = -2x. The output for each x is calculated by multiplying x by -2. The results are f(x) = 4, 2, 0, -2, and -4 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = -3x. This involves creating a table of values for x and f(x) using f(x) = -3x. The output for each x is calculated by multiplying x by -3. The results are f(x) = 6, 3, 0, -3, and -6 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = -4x. This involves creating a table of values for x and f(x) using f(x) = -4x. The output for each x is calculated by multiplying x by -4. The results are f(x) = 8, 4, 0, -4, and -8 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = -5x. This involves creating a table of values for x and f(x) using f(x) = -5x. The output for each x is calculated by multiplying x by -5. The results are f(x) = 10, 5, 0, -5, and -10 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

Find the output values for the input values -2, -1, 0, 1, 2 using a function machine with the rule f(x) = 2x + 1. This involves creating a table of values for x and f(x) using f(x) = 2x + 1. The output for each x is calculated by doubling x and then adding 1. The results are f(x) = -3, -1, 1, 3, and 5 for x = -2, -1, 0, 1, and 2, respectively.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = 2x + 4. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, 4), (1, 6), (2, 8), (3, 10), and (4, 12), illustrating how the y-value increases by 2 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = -x + 6. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, 6), (1, 5), (2, 4), (3, 3), and (4, 2), illustrating how the y-value decreases by 1 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = -x - 8. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, -8), (1, -9), (2, -10), (3, -11), and (4, -12), showing how the y-value decreases by 1 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the linear function y = -x. The image displays both a graph and a table representing this function. The table includes coordinate pairs (0, 0), (1, -1), (2, -2), (3, -3), and (4, -4), illustrating how the y-value decreases by 1 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = 0.5x + 4. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 4), (1, 4.5), (2, 5), (3, 5.5), and (4, 6), showing how the y-value increases by 0.5 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = 0.1x - 5. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, -5), (1, -4.9), (2, -4.8), (3, -4.7), and (4, -4.6), illustrating how the y-value increases by 0.1 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = 0.2x. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 0), (1, 0.2), (2, 0.4), (3, 0.6), and (4, 0.8), showing how the y-value increases by 0.2 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = -0.5x + 7. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, 7), (1, 6.5), (2, 6), (3, 5.5), and (4, 5), illustrating how the y-value decreases by 0.5 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = -0.1x - 4. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, -4), (1, -4.1), (2, -4.2), (3, -4.3), and (4, -4.4), showing how the y-value decreases by 0.1 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = -0.6x. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, 0), (1, -0.6), (2, -1.2), (3, -1.8), and (4, -2.4), illustrating how the y-value decreases by 0.6 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = 0x. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 0), (1, 0), (2, 0), (3, 0), and (4, 0), showing how the y-value remains constant at 0 for all values of x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = 3x - 2. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, -2), (1, 1), (2, 4), (3, 7), and (4, 10), showing how the y-value increases by 3 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = 0x - 6. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, -6), (1, -6), (2, -6), (3, -6), and (4, -6), illustrating how the y-value remains constant at -6 for all values of x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = 0 * x + 4. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 4), (1, 4), (2, 4), (3, 4), and (4, 4), showing how the y-value remains constant at 4 for all values of x.

Topic

Linear Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the linear function y = 5x. The image displays both a graph and a table representing this function. The table includes coordinate pairs (0, 0), (1, 5), (2, 10), (3, 15), and (4, 20), illustrating how the y-value increases by 5 for each unit increase in x.

Topic

Linear Functions

Description

This example showcases the process of creating a table of x-y coordinates and graphing the linear function y = -2x + 5. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 5), (1, 3), (2, 1), (3, -1), and (4, -3), demonstrating how the y-value decreases by 2 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the creation of a table of x-y coordinates and the graphing of the linear function y = -3x - 4. The image displays both a graph and a table representing this function. The table includes coordinate pairs (0, -4), (1, -7), (2, -10), (3, -13), and (4, -16), showing how the y-value decreases by 3 for each unit increase in x.

Topic

Linear Functions

Description

This example demonstrates how to create a table of x-y coordinates and graph the linear function y = -6x. The image shows both a graph and a table representing this function. The table includes coordinate pairs (0, 0), (1, -6), (2, -12), (3, -18), and (4, -24), illustrating how the y-value decreases by 6 for each unit increase in x.

Topic

Linear Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the linear function y = x + 7. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 7), (1, 8), (2, 9), (3, 10), and (4, 11), showing how the y-value increases by 1 for each unit increase in x.