Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Value Function
The absolute value function, denoted as f(x) = |x|, outputs the non-negative value of x. This function has a V-shaped graph that opens upwards, with its vertex at the origin (0,0). Understanding this function is crucial as it serves as the foundation for applying transformations to graph other functions.
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Transformations of Functions
Transformations involve shifting, reflecting, stretching, or compressing the graph of a function. For example, adding a constant inside the absolute value affects horizontal shifts, while adding outside affects vertical shifts. In the function g(x) = -|x + 4| + 2, the transformations include a horizontal shift left by 4 units, a reflection across the x-axis, and a vertical shift up by 2 units.
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Domain & Range of Transformed Functions
Graphing Techniques
Graphing techniques involve plotting points and understanding how transformations affect the shape and position of a graph. For the function g(x), one must first graph f(x) = |x|, then apply the identified transformations step-by-step. This systematic approach helps visualize the final graph accurately and understand the relationship between the original and transformed functions.
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Graphs and Coordinates - Example