Function transformations in calculus involve altering a function's graph through translations, dilations, and reflections. These modifications can be vertical or horizontal, affecting the y-coordinates or x-coordinates, respectively. Understanding these changes is crucial for graphing functions accurately, and this knowledge applies to various function families, from exponential to polynomial.
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Function transformations in calculus are operations that alter the graph of a function while preserving its overall shape, and can be classified as either horizontal or vertical
Translations
Translations, or shifts, involve adding or subtracting a constant to the input variable or output variable of a function
Dilations
Dilations, or stretches and compressions, involve multiplying the input variable or output variable of a function by a constant
Reflections
Reflections involve multiplying the function or its input by -1, resulting in a mirror image of the graph across the corresponding axis
Misunderstandings can occur with horizontal transformations, highlighting the importance of a solid conceptual grasp of function transformations
Vertical transformations involve adding or subtracting a constant to the function or multiplying the function by a constant
Horizontal transformations involve adding or subtracting a constant to the input variable or multiplying the input variable by a constant
Reflections involve multiplying the function or its input by -1
The order in which transformations are applied to a function can significantly influence the final appearance of the graph
When combining multiple transformations of the same category, the order of application is critical, but it does not impact the resulting graph when transformations are of different categories
To find the new position of points on a graph after a transformation, one must apply the inverse of the transformation to the original coordinates
Function transformations can be applied to all types of functions, including exponential, logarithmic, polynomial, and rational functions
Each function family has a general transformation formula that incorporates parameters for vertical and horizontal translations, dilations, and reflections
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Vertical Translation Effect
Adding/subtracting constant c to f(x) moves graph up/down by c units.
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Vertical Dilation Factors
Multiplying f(x) by constant c > 1 stretches graph vertically; 0 < c < 1 compresses it.
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Reflection Transformation
Multiplying f(x) or x by -1 reflects graph across y-axis or x-axis, respectively.
