There Are Many Ways To Solve Math Problems

There Are Many Ways To Go About Solving Math Problems For This Assign

For this assignment, two functions will be analyzed to understand their graphs, key points, and transformations. The functions are y = -¾x + 4 and y = 2 |x|, where |x| denotes the absolute value of x. The task involves plotting points, identifying key features such as intercepts and vertices, discussing the general shape and position of each graph, establishing the domain and range in interval notation, determining whether each is a function, and then applying a translation to one of the graphs to observe how it affects the equation through a transformation while incorporating the function, relation, domain, and range.

Graphing and Analyzing y = -¾x + 4

To graph y = -¾x + 4, I first selected five integer points by choosing x-values and calculating corresponding y-values:

• x = 0: y = -¾(0) + 4 = 4
• x = 4: y = -¾(4) + 4 = -3 + 4 = 1
• x = -4: y = -¾(-4) + 4 = 3 + 4 = 7
• x = 8: y = -¾(8) + 4 = -6 + 4 = -2
• x = -8: y = -¾(-8) + 4 = 6 + 4 = 10

Key points include the y-intercept at (0, 4), which is also the starting point of the line. The graph is a straight line with a negative slope of -¾, indicating it descends from left to right. The line passes through the points (4, 1), (-4, 7), (8, -2), and (-8, 10). The shape is linear, extending infinitely in both directions.

The domain of this function is all real numbers, expressed as (-∞, ∞). The range is all real numbers less than or equal to 4, since the line extends downward without bound as x increases, but the highest y-value at x=0 is 4. So, the range is (-∞, 4].

This function is indeed a function because each x-value has exactly one y-value, satisfying the definition of a function.

Graphing and Analyzing y = 2|x|

The absolute value function y = 2|x| is V-shaped, symmetric about the y-axis. To plot, I choose five x-values:

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

The key point is the vertex at (0, 0), which is the minimum point of the graph. The graph increases linearly on both sides of the vertex, with slopes of 2 and -2, forming a "V" shape. The general shape is symmetric about the y-axis.

The domain is all real numbers, (-∞, ∞), since x can be any real number. The range is y ≥ 0, which in interval notation is [0, ∞), because the output y-values are never negative.

This is a function because for each x-value, there is only one y-value, even though the graph is not a line but an absolute value curve.

Effect of Transformation on y = 2|x|

Next, I select the absolute value function graph and apply a shift: three units upward and four units to the left. The transformed graph will be affected by these transformations. Moving the graph three units upward increases every y-value by 3, so the equation becomes:

y = 2|x| + 3

Moving the graph four units to the left involves replacing x with (x + 4). This horizontal shift means the graph shifts leftward, effectively changing the input to x + 4. The new equation becomes:

y = 2|x + 4| + 3

This rewritten equation encapsulates both transformations: the absolute value function is shifted four units to the left (by replacing x with x + 4), and then shifted three units upward by adding +3 to the whole expression. These changes modify the relation by repositioning the graph without affecting its fundamental shape, which remains the same.

From the perspective of the domain, the original domain (-∞, ∞) remains unchanged because shifts do not restrict x-values, so the domain of the transformed equation is still (-∞, ∞). The range, however, changes; initially y ≥ 0, but after shifting upward by 3, the new range is y ≥ 3, or [3, ∞).

This transformation demonstrates how manipulating the parameters of an equation can produce a new relation with the same general shape, but positioned differently within a coordinate plane.

Conclusion

Understanding the properties of different functions, including linear and absolute value functions, involves analyzing their equations, plotting key points, and examining their shape and position. Identifying whether a graph represents a function depends on whether each x-value corresponds to exactly one y-value. The process of applying transformations such as shifting graphs horizontally or vertically showcases the flexibility in manipulating functions while maintaining their core properties. Recognizing how these adjustments affect the relation—specifically the domain and range—is essential in understanding the behavior of various functions in mathematical analysis. Such insights aid in solving more complex problems and deepen comprehension of the functional relationships shaping the graphical representations.

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