뒤로Transformations of Functions: Stretching, Shrinking, and Vertical Translation
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Transformations of Functions
Stretching and Shrinking of Functions
Transformations such as stretching and shrinking change the shape of a function's graph by multiplying the function by a constant. These transformations are important in understanding how graphs behave under scaling.
Vertical Stretch: If , the graph of is stretched vertically by a factor of .
Vertical Shrink: If , the graph of is shrunk vertically by a factor of .
Reflection and Stretch/Shrink: If , the graph is reflected across the x-axis and also stretched or shrunk depending on the absolute value of .
Example: Consider the functions , , and .
: The basic square root function.
: The graph is stretched vertically by a factor of 2.
: The graph is shrunk vertically by a factor of .
Graphical Comparison: The graph shows how each transformation affects the shape and position of the square root function. For example, rises more steeply than , while rises more slowly.
Vertical Translation
A vertical translation shifts the graph of a function up or down without changing its shape. This is achieved by adding or subtracting a constant to the function.
Definition: If , then the graph of is a translation of units upward. If , then the graph of is a translation of units downward.
Example: Graph , , and .
: The basic absolute value function.
: The graph is shifted 2 units upward.
: The graph is shifted 2 units downward.
Summary Table: Effects of Vertical Stretch/Shrink and Translation
Transformation | Equation | Effect on Graph |
|---|---|---|
Vertical Stretch | , | Stretches graph vertically by factor |
Vertical Shrink | , | Shrinks graph vertically by factor |
Vertical Translation Up | , | Shifts graph up by units |
Vertical Translation Down | , | Shifts graph down by units |
Additional info: These transformations are foundational in College Algebra and are used to analyze and graph a wide variety of functions, including polynomial, radical, and absolute value functions.
