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Given the equations y = -3x + 4 and y = 2|x|, the objective is to graph these functions, analyze their key features, and discuss their transformations comprehensively.

First, we will generate five points for each equation, including their intercepts and notable points, to help plot and understand the shape and position of each graph.

Graphing and Analyzing the Equations

Equation 1: y = -3x + 4

This is a linear function with a slope of -3 and a y-intercept at (0, 4). The slope indicates that for every 1 unit increase in x, y decreases by 3 units.

Five points for y = -3x + 4:

• When x = 0: y = -3(0) + 4 = 4 → (0, 4)
• When x = 1: y = -3(1) + 4 = 1 → (1, 1)
• When x = 2: y = -3(2) + 4 = -2 → (2, -2)
• When x = -1: y = -3(-1) + 4 = 7 → (-1, 7)
• When x = -2: y = -3(-2) + 4 = 10 → (-2, 10)

Key points include the y-intercept (0, 4). The graph is a straight line decreasing from left to right, extending infinitely in both directions.

Domain: All real numbers, in interval notation: (-∞, ∞)

Range: All real numbers, in interval notation: (-∞, ∞)

This equation is a function because each x-value has exactly one y-value, obeying the vertical line test.

Equation 2: y = 2|x|

This is an absolute value function, which forms a V-shape symmetrical about the y-axis with the vertex at (0, 0).

Five points for y = 2|x|:

• When x = 0: y = 2|0| = 0 → (0, 0)
• When x = 1: y = 2|1| = 2 → (1, 2)
• When x = -1: y = 2|-1| = 2 → (-1, 2)
• When x = 2: y = 2|2| = 4 → (2, 4)
• When x = -2: y = 2|-2|= 4 → (-2, 4)

The graph opens upward, with the vertex at the origin. It is symmetric across the y-axis.

Domain: All real numbers, (-∞, ∞)

Range: [0, ∞)

This is a function because each x-value produces exactly one y-value, satisfying the vertical line test.

Discussion of the Graphs and Transformations

General shape and location

The graph of y = -3x + 4 is a straight line with a negative slope, positioned so that it crosses the y-axis at 4 and slants downward as x increases. The steep slope indicates rapid change in y with respect to x.

The graph of y = 2|x| is a V-shaped graph centered at the origin, opening upwards. Its shape is determined by the absolute value, which creates symmetry and a sharp vertex.

Key points and intercepts

For y = -3x + 4, the y-intercept at (0, 4) is an important key point. For x-intercept: set y=0; 0 = -3x + 4 → x = 4/3.

For y= 2|x|, the vertex at (0,0) is the key point, with intercepts at (0,0) and symmetric points at (±1, 2) and (±2, 4).

Effect of transformations: shift upward and to the left

Choosing y = -3x + 4 as our graph, suppose we shift this graph three units upward and four units to the left. This transformation involves a vertical shift of +3 and a horizontal shift of -4.

Applying these transformations to the original equation, the transformed function becomes:

 y = -3(x + 4) + (4 + 3) 

This simplifies to:

 y = -3(x + 4) + 7 

In this form, the graph has moved four units to the left, as indicated by (x + 4), and three units upward, as indicated by +7.

In discussing how these transformations affect the function / relation, note that the vertical line test still confirms the transformed equations as functions. The transformation does not alter the fundamental shape but changes the position of the graph in the coordinate plane. Therefore, the shape remains a straight line, and the general location shifts accordingly.

Impact of Transformation on the Function/Relation and Vertical Line Test

The transformation preserves the function relation property because for every x-value, there's still exactly one y-value. The vertical line test confirms the graph remains a function after the shift. The transformation illustrates the concept of applying a translation to the original function, which shifts the entire relation without changing its fundamental characteristics.

Conclusion

In summary, the two functions examined—the linear function y = -3x + 4 and the absolute value function y = 2|x|—each have distinctive shapes and key features. Understanding their key points, domain, range, and the effects of transformations provides a comprehensive view of their behavior and graph characteristics. The shifted transformations demonstrate how shifting functions horizontally and vertically affect their equations, reinforcing the concepts of functions and relations within algebra and coordinate geometry.

References

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