Illustrative Math Alignment: Grade 8 Unit 3

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.

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 - 8. The image displays both a graph and a table representing this function. The table includes coordinate pairs (0, -8), (1, -7), (2, -6), (3, -5), and (4, -4), illustrating how the y-value increases 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. The image presents both a graph and a table for this function. The table includes coordinate pairs (0, 0), (1, 1), (2, 2), (3, 3), and (4, 4), showing how the y-value increases by 1 for each unit increase in x.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 28x + 7y = 35 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = -4x + 5 by isolating y. The y-intercept is found as 5 (where x = 0), and the x-intercept is calculated as 1.25 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 6x - 2y = 6 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = 3x - 3. The y-intercept is -3 (where x = 0), and the x-intercept is 1 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 24x + 12y = 48 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = -2x + 4. The y-intercept is 4 (where x = 0), and the x-intercept is 2 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 15x + 3y = 15 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = -5x + 5. The y-intercept is 5 (where x = 0), and the x-intercept is 1 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 12x + 3y = 21 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = -4x + 7. The y-intercept is 7 (where x = 0), and the x-intercept is 1.75 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 10x + 5y = 15 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = -2x + 3. The y-intercept is 3 (where x = 0), and the x-intercept is 1.5 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 10x - 2y = 10 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = 5x - 5. The y-intercept is -5 (where x = 0), and the x-intercept is 1 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 32x - 8y = 32 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = 4x - 4. The y-intercept is -4 (where x = 0), and the x-intercept is 1 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 16x - 8y = 40 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = 2x - 5. The y-intercept is -5 (where x = 0), and the x-intercept is 2.5 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

Find the x- and y-intercepts for the linear equation 20x - 10y = 60 given in standard form. Convert it to slope-intercept form.

The equation is rewritten as y = 2x - 6. The y-intercept is -6 (where x = 0), and the x-intercept is 3 (where y = 0). These intercepts are plotted on a graph.

Linear functions are a fundamental concept in algebra, where we explore the relationships between variables. The examples in this collection highlight how to find intercepts and analyze the relationships they represent in a linear context.

Topic

Linear Functions

Description

This example demonstrates how to find the equation of a line passing through two given points: (6, 4) and (8, 8). The slope is calculated using the formula (y2 - y1) / (x2 - x1), resulting in a slope of 2. Using the point-slope form of a line, y - y1 = m(x - x1), the equation is derived as y - 8 = 2(x - 8), which simplifies to y = 2x - 8.

Topic

Linear Functions

Description

The image shows a coordinate plane with two points (-6, -2) and (-2, -6) marked. It provides a step-by-step solution to find the equation of the line passing through these points. The slope is calculated, and the equation is derived using point-slope form. The slope is calculated as (y2 - y1) / (x2 - x1) = (-6 - (-2)) / (-2 - (-6)) = -4 / 4 = -1. Using point-slope form, y - y1 = m(x - x1), the equation is derived as y + 2 = -(x + 6), which simplifies to y = -x - 8.

Topic

Linear Functions

Description

The image shows a coordinate plane with two points (-8, -4) and (-2, -4) marked. It provides a step-by-step solution to find the equation of the line passing through these points. The slope is calculated as zero, indicating a horizontal line. The slope is calculated as (y2 - y1) / (x2 - x1) = (-4 - (-4)) / (-8 - (-2)) = 0 / -6 = 0. Since the slope is zero, it indicates a horizontal line at y = -4. Using point-slope form, the equation becomes y + 4 = 0, which simplifies to y = -4.

Topic

Linear Functions

Description

The image shows a coordinate plane with two points (-5, -8) and (-5, -3) marked. It provides a step-by-step solution to find the equation of the line passing through these points. The slope is undefined, indicating a vertical line. The slope is calculated as (y2 - y1) / (x2 - x1) = (-3 - (-8)) / (-5 - (-5)) = 5 / 0, which is undefined. Since the slope is undefined, it indicates a vertical line at x = -5. Therefore, the equation of the line is simply x = -5.

Topic

Linear Functions

Description

This image shows a graph with two points (2, -4) and (6, -2) marked. The example demonstrates how to find the equation of a line passing through these points. The slope is calculated as 1/2, and the point-slope form is used to derive the equation of the line. The slope formula is used: (y2 - y1) / (x2 - x1) = (-2 - (-4)) / (6 - 2) = 2 / 4 = 1/2. Then, using the point-slope form: y - (-2) = (1/2)(x - 6), which simplifies to y = (1/2)x - 5.

Topic

Linear Functions

Description

This image shows a graph with two points (2, -3) and (5, -6). The example demonstrates how to find the equation of a line passing through these points. The slope is calculated as -1, and the point-slope form is used to derive the equation. The slope formula is used: (y2 - y1) / (x2 - x1) = (-3 - (-6)) / (2 - 5) = 3 / -3 = -1. Then, using the point-slope form: y - (-3) = -(x - 2), which simplifies to y = -x - 1.

Topic

Linear Functions

Description

This image shows a graph with two points (3, -4) and (7, -4). The example demonstrates how to find the equation of a horizontal line passing through these points. The slope is calculated as 0, and the point-slope form is used to derive the equation. The slope formula is used: (y2 - y1) / (x2 - x1) = (-4 - (-4)) / (7 - 3) = 0 / 4 = 0. Then, using the point-slope form: y - (-4) = 0(x - 3), which simplifies to y = -4.