Here are the essential concepts you must grasp in order to answer the question correctly.
Function Transformation
Function transformation refers to the changes made to the graph of a function based on modifications to its equation. These transformations include shifts, stretches, compressions, and reflections. Understanding how to manipulate the function's equation allows one to predict how the graph will change, which is essential for graphing functions like g(x) based on f(x).
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Horizontal Shift
A horizontal shift occurs when the input of a function is altered, resulting in the graph moving left or right. In the function g(x) = 2f(x+2) − 1, the term (x+2) indicates a shift of the graph of f(x) to the left by 2 units. This concept is crucial for accurately positioning the graph of g(x) relative to f(x).
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Vertical Stretch and Shift
Vertical stretch and shift involve scaling the output of a function and moving it up or down. In g(x) = 2f(x+2) − 1, the coefficient '2' indicates a vertical stretch by a factor of 2, making the graph of f(x) taller. The '−1' indicates a downward shift of the entire graph by 1 unit. Understanding these transformations is key to accurately graphing g(x).
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