Illustrative Math Alignment: Grade 8 Unit 3

Topic

Exponential Functions

Description

This example demonstrates the creation of a table of x-y coordinates and the graphing of the exponential function y = -3 * 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 larger magnitude.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = -1 * 2-3x. 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 factor in the 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 = -4 * 2-3x. 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 negative coefficient, a larger magnitude, and a factor in the exponent.

Topic

Exponential Functions

Description

This example illustrates the process of creating a table of x-y coordinates and graphing the exponential function y = -0.5 * 2x. The image presents both a table of values and a graph, displaying the function's behavior. The table lists x-y coordinates for x values from -2 to 2, with corresponding y values calculated using the given function. This representation helps students visualize an exponential function with a negative fractional coefficient.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x + a) from f(x) = x, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x shifts horizontally by adding a to the input. substituting a = 1, the new function becomes f(x + a) = x + 1. the graph shifts one unit up.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x) + a from f(x) = x2, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x2 shifts vertically by adding a to the output. substituting a = 1, the new function becomes f(x) + a = x2 + 1. the graph shifts one unit upward.

Topic

Arithmetic

Description

The problem involves analyzing the effect on the function f(x) = x2 when a constant a is subtracted. The specific scenario is for a = 1. The goal is to determine how the graph of f(x) is transformed. This example shows the effect on the function when subtracting a constant a from f(x) shifts the graph of f(x) downward by a units. for a = 1, the function becomes f(x) - a = x2 - 1. this results in a vertical shift down, as shown by the new graph in blue compared to the original in red.

Topic

Arithmetic

Description

The problem examines the effect on the function f(x) = x2 when it is scaled by a constant a. The specific case here is a = -1. The goal is to describe the transformation caused by the scaling. This example shows the effect on the function when multiplying f(x) by a constant a reflects the graph across the x-axis if a < 0. for a = -1, the function becomes a * f(x) = -x2. this results in a reflection of the original graph (in red) to the inverted graph (in blue).

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x - a) from f(x) = x, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x shifts horizontally by subtracting a from the input. substituting a = 1, the new function becomes f(x - a) = x - 1. the graph shifts one unit down.

Topic

Arithmetic

Description

Analyze the effect of building a new function a * f(x) from f(x) = x, where a = 2. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x scales vertically by multiplying the output by a. substituting a = 2, the new function becomes a * f(x) = 2x. the graph is stretched vertically.

Topic

Arithmetic

Description

Analyze the effect of building a new function a * f(x) from f(x) = x, where a = -2. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x scales vertically and reflects across the x-axis due to the negative value of a. substituting a = -2, the new function becomes a * f(x) = -2x. the graph is reflected and stretched vertically.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x + a) from f(x) = 3x, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = 3x shifts horizontally by adding a to the input. substituting a = 1, the new function becomes f(x + a) = 3x + 1. the graph shifts one unit up.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x - a) from f(x) = 3x, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = 3x shifts horizontally by subtracting a from the input. substituting a = 1, the new function becomes f(x - a) = 3x - 1. the graph shifts one unit down.

Topic

Arithmetic

Description

Analyze the effect of building a new function a * f(x + b) from f(x) = 3x, where a = 2 and b = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = 3x first shifts horizontally by adding b to the input, and then scales vertically by a. substituting a = 2 and b = 1, the new function becomes a * f(x + b) = 6x + 2. the graph is shifted and stretched vertically.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x + a) from f(x) = x2, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x2 shifts horizontally by adding a to the input. substituting a = 1, the new function becomes f(x + a) = (x + 1)2. the graph shifts one unit to the left.

Topic

Arithmetic

Description

Analyze the effect of building a new function f(x - a) from f(x) = x2, where a = 1. Graph the transformation and calculate the new function. This example shows the effect on the function when the function f(x) = x2 shifts horizontally by subtracting a from the input. substituting a = 1, the new function becomes f(x - a) = (x - 1)2. the graph shifts one unit to the right.

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

Quadratics

Description

This example illustrates the relationship between quadratic functions and their graphical representations. The image shows a table of values and the corresponding graph of a quadratic function, highlighting how changes in the equation affect the graph's shape and position. Understanding this relationship is crucial for interpreting and predicting the behavior of quadratic functions. Skills involved include plotting points, recognizing parabolic shapes, and analyzing vertex and axis of symmetry.

Topic

Quadratics

Topic

Quadratics

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 highlights the relationship between the equation and its graph, showing the parabolic curve and how it opens downward. Key features such as the vertex and intercepts are easily identifiable, which are crucial for understanding the function's behavior.

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. The example illustrates how the coefficients in the quadratic equation affect the graph's shape and direction, allowing for a deeper understanding of the function's properties.

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 the symmetry of the parabola and how the vertex acts as a turning point, which is critical for graph analysis and interpretation.

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. The graph clearly shows the parabola's opening direction and the impact of the quadratic term, aiding in the comprehension of graph transformations.

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. The example demonstrates how the vertex form of the equation can be used to predict the graph's position and shape, enhancing understanding of vertex transformations.

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. The graphic representation is useful for identifying roots and how they correspond to the x-intercepts of the graph, which is fundamental for solving quadratic equations graphically.

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. The graphical representation is useful for determining the kind of discriminant and the number of roots, which is visually represented by the points where the graph intersects the x-axis.

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. These representations are useful to show the axis of symmetry can be used to find the vertex and how this symmetry is reflected in the graph.

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

Topic

Quadratics

Description

This example demonstrates the concept of quadratic functions through a table of values and its graph. It emphasizes how the quadratic function translates into a parabolic graph, showcasing the vertex and symmetry. This visual representation aids in comprehending the impact of coefficients on the graph's curvature and direction. Mastery of this concept involves skills in graphing, interpreting quadratic equations, and identifying key features such as intercepts and vertex.

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 and you can see the relationship between the parameters of the function and how they are displayed graphically.

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