Illustrative Math Alignment: Grade 8 Unit 3

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

Topic

Radical Functions

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Radical Functions

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Radical Functions

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Special Functions

Topic

Special Functions

Description

This example presents a step function defined by y = floor(-0.5x + 1). The graph shows horizontal segments stepping down as x increases. The table contains x values (-4, -2, 0, 2, 4) and their corresponding y values (3, 2, 1, 0, -1), illustrating how the function behaves at these specific points.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(2x) + 1. The table shows x values (-4, -2, 0, 2, 4) and their corresponding y values (-7, -3, 1, 5, 9). The graph displays horizontal steps increasing as x increases, demonstrating how the function behaves across different intervals.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(-2x) + 1. The table shows x values (-4, -2, 0, 2, 4) and their corresponding y values (9, 5, 1, -3, -7). The graph displays horizontal line segments stepping down as x increases, illustrating how the function behaves across different intervals.

Topic

Special Functions

Description

This example showcases a step function defined by y = floor(-2x) + 1. The graph features red horizontal segments and black points at key coordinates, with x values of -4, -2, 0, 2, and 4, and corresponding y values of 9, 5, 1, -3, and -7. This visual representation helps students understand how the function behaves across different intervals.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(0.5x) + 1. The graph features red horizontal segments and black points at key coordinates, with x values of -4, -2, 0, 2, and 4, and corresponding y values of -1, 0, 1, 2, and 3. This visual representation helps students understand how the function behaves when the coefficient of x is a fraction.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(-0.5x) + 1. The graph features red horizontal segments and black points at key coordinates, with x values of -4, -2, 0, 2, and 4, and corresponding y values of 3, 2, 1, 0, and -1. This visual representation helps students understand how the function behaves when the coefficient of x is a negative fraction.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(x + 1) + 1. The graph features red horizontal segments and black points at key coordinates, with x values of -4, -2, 0, 2, and 4, and corresponding y values of -2, 0, 2, 4, and 6. This visual representation helps students understand how the function behaves across different intervals.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(2x + 1) + 1. The graph is plotted with points (-4, -6), (-2, -2), (0, 2), (2, 6), and (4, 10), showing horizontal steps that increase as x increases. This visual representation helps students understand how the function behaves when there's a coefficient greater than 1 inside the floor function.

Topic

Special Functions

Description

This example showcases a step function defined by y = floor(-2x + 1) + 1. The graph consists of horizontal steps decreasing sharply as x increases, with points plotted at (-4, 10), (-2, 6), (0, 2), (2, -2), and (4, -6). This visual representation helps students understand how the function behaves when there's a negative coefficient inside the floor function.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(0.5x + 1) + 1. The graph consists of horizontal steps increasing gradually as x increases, with points plotted at (-4, 0), (-2, 1), (0, 2), (2, 3), and (4, 4). This visual representation helps students understand how the function behaves when there's a fractional coefficient inside the floor function.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(2x). The table presents x values (-4, -2, 0, 2, 4) and corresponding y values (-8, -4, 0, 4, 8). The graph shows how this function behaves, with distinct horizontal segments that illustrate the "stepping" nature of the function. This example helps students visualize how multiplying x by 2 inside the floor function affects the graph's appearance.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(-0.5x + 1) + 1. The graph consists of horizontal steps decreasing from left to right, with points plotted at (-4, 4), (-2, 3), (0, 2), (2, 1), and (4, 0). This visual representation helps students understand how the function behaves when there's a negative fractional coefficient inside the floor function.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(x) - 1. The graph features horizontal steps at each integer value of x, with points plotted at (-4, -5), (-2, -3), (0, -1), (2, 1), and (4, 3). This visual representation helps students understand how the function behaves when a constant is subtracted from the floor function.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(2x) - 1. The graph features horizontal steps that increase as x increases, with points plotted at (-4, -9), (-2, -5), (0, -1), (2, 3), and (4, 7). This visual representation helps students understand how the function behaves when the input is multiplied by 2 and a constant is subtracted from the floor function.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(-2x) - 1. The graph features horizontal steps that decrease as x increases, with points plotted at (-4, 7), (-2, 3), (0, -1), (2, -5), and (4, -9). This visual representation helps students understand how the function behaves when the input is multiplied by a negative number and a constant is subtracted from the floor function.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(0.5x) - 1. The graph features horizontal steps that increase slowly as x increases, with points plotted at (-4, -3), (-2, -2), (0, -1), (2, 0), and (4, 1). This visual representation helps students understand how the function behaves when the input is multiplied by a fraction and a constant is subtracted from the floor function.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(-0.5x) - 1. The graph is a series of horizontal steps, with points plotted at (-4, 1), (-2, 0), (0, -1), (2, -2), and (4, -3). This visual representation helps students understand how the function behaves when the input is multiplied by a negative fraction and a constant is subtracted from the floor function.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(x + 1) - 1. The graph consists of horizontal steps, with points plotted at (-4, -4), (-2, -2), (0, 0), (2, 2), and (4, 4). This visual representation helps students understand how the function behaves when a constant is added inside the floor function and another constant is subtracted outside.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(2x + 1) - 1. The graph features horizontal steps with points plotted at (-4, -8), (-2, -4), (0, 0), (2, 4), and (4, 8). This visual representation helps students understand how multiplying x by a coefficient greater than one affects the frequency of steps while subtracting a constant shifts them vertically.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(-2x + 1) - 1. The graph consists of horizontal steps with points plotted at (-4, 8), (-2, 4), (0, 0), (2, -4), and (4, -8). This visual representation helps students understand how the function behaves when the input is multiplied by a negative number, a constant is added inside the floor function, and another constant is subtracted outside.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(0.5x + 1) - 1. The graph features horizontal steps with key points labeled, and x values (-4, -2, 0, 2, 4) corresponding to y values (-2, -1, 0, 1, 2). This visual representation helps students understand how the function behaves when the input is multiplied by a fraction, a constant is added inside the floor function, and another constant is subtracted outside.

Topic

Special Functions

Description

This example showcases a step function defined by y = floor(-2x). The table lists x values (-4, -2, 0, 2, 4) and corresponding y values (8, 4, 0, -4, -8). The graph illustrates how this function behaves, with distinct horizontal segments that demonstrate the "stepping" nature of the function. This example helps students visualize how multiplying x by -2 inside the floor function affects the graph's appearance and direction.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(-0.5x + 1) - 1. The graph consists of horizontal steps with labeled points, and x values (-4, -2, 0, 2, 4) corresponding to y values (2, 1, 0, -1, -2). This visual representation helps students understand how the function behaves when the input is multiplied by a negative fraction, a constant is added inside the floor function, and another constant is subtracted outside.

Topic

Special Functions

Description

This example illustrates a step function defined by y = floor(x) - 1. The graph consists of horizontal steps with points plotted at (-4, -5), (-2, -3), (0, -1), (2, 1), and (4, 3). This visual representation helps students understand how subtracting a constant from the floor function shifts the graph vertically.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = floor(2x) - 1. The graph features horizontal steps with points plotted at (-4, -9), (-2, -5), (0, -1), (2, 3), and (4, 7). This visual representation helps students understand how multiplying x by two affects the frequency of steps while subtracting one shifts them vertically.

Topic

Special Functions

Description

This example illustrates a step function defined by y = -floor(-2x) - 1. The graph displays horizontal segments at different integer values of x, with points plotted at (-4, -9), (-2, -5), (0, -1), (2, 3), and (4, 7). This visual representation helps students understand how the function behaves when the input is multiplied by a negative number, the floor function is negated, and a constant is subtracted.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = -floor(0.5x) - 1. The graph shows horizontal steps at different x-values, with points plotted at (-4, 1), (-2, 0), (0, -1), (2, -2), and (4, -3). This visual representation helps students understand how the function behaves when the input is multiplied by a fraction, the floor function is negated, and a constant is subtracted.

This is part of a collection of math examples that focus on special functions. This includes step functions and radical functions.

Topic

Special Functions

Description

This example illustrates a step function defined by y = -floor(x + 1) - 1. The graph consists of horizontal steps with points plotted at (-4, 2), (-2, 0), (0, -2), (2, -4), and (4, -6). This visual representation helps students understand how negating the floor function and adding constants both inside and outside the function affects the graph's appearance.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = -floor(2x + 1) - 1. The graph features distinct horizontal steps, with x-values from -4 to 4 and corresponding y-values of 6, 2, -2, -6, and -10. This visual representation helps students understand how multiplying x by 2, adding a constant inside the floor function, and then negating and subtracting outside affects the graph's appearance.

Topic

Special Functions

Description

This example illustrates a step function defined by y = -floor(-2x + 1) - 1. The graph consists of horizontal steps, with x-values from -4 to 4 and corresponding y-values of -10, -6, -2, 2, and 6. This visual representation helps students understand how negating x inside the floor function, adding a constant, and then negating the entire function and subtracting affects the graph's appearance.

Topic

Special Functions

Description

This example demonstrates a step function defined by y = -floor(0.5x + 1) - 1. The graph features horizontal steps at various points, with x-values from -4 to 4 and corresponding y-values of 0, -1, -2, -3, and -4. This visual representation helps students understand how multiplying x by a fraction inside the floor function, adding a constant, and then negating and subtracting outside affects the graph's appearance.

Topic

Special Functions

Description

This example presents a step function defined by y = floor(0.5x). The table includes x values (-4, -2, 0, 2, 4) and corresponding y values (-2, -1, 0, 1, 2). The graph illustrates how this function behaves, with distinct horizontal segments that demonstrate the "stepping" nature of the function. This example helps students visualize how multiplying x by 0.5 inside the floor function affects the graph's appearance, particularly the width of the steps.