A transformed function is a function that has been altered from its original form through various operations, such as shifting, stretching, compressing, or reflecting. These transformations modify the graph of the function while maintaining its fundamental characteristics. Understanding transformed functions is crucial for analyzing and predicting the behavior of functions in different contexts.
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A vertical translation of a function can be achieved by adding or subtracting a constant from the function's output (e.g., $$f(x) + c$$).
A horizontal translation occurs by adding or subtracting a constant from the input (e.g., $$f(x - c)$$), shifting the graph left or right.
Reflection over the x-axis is done by multiplying the function by -1 (e.g., $$-f(x)$$), which flips the graph upside down.
Stretching a graph vertically can be accomplished by multiplying the function by a factor greater than 1 (e.g., $$af(x)$$ where $$a > 1$$), while compressing it is done with a factor between 0 and 1.
Horizontal stretching and compressing are achieved by multiplying the input by a factor (e.g., $$f(bx)$$ where $$b > 1$$ compresses and $$0 < b < 1$$ stretches).
Review Questions
How do translations affect the positioning of a transformed function's graph?
Translations affect the positioning of a transformed function's graph by shifting it either horizontally or vertically without altering its shape. A vertical translation occurs when a constant is added to or subtracted from the output of the function, moving it up or down. A horizontal translation involves adjusting the input by adding or subtracting a constant, moving the graph left or right. Both types of translations maintain the original structure of the function but change where it appears on the coordinate plane.
Describe how reflections transform a function's graph and provide examples.
Reflections transform a function's graph by flipping it over an axis. For instance, reflecting a function over the x-axis is done by multiplying the entire function by -1, resulting in $$-f(x)$$, which inverts all y-values. Reflecting over the y-axis involves substituting $$x$$ with $$-x$$ in the function, resulting in $$f(-x)$$. These transformations change how the graph appears visually while retaining key features such as intercepts.
Evaluate how stretching and compressing affects both the visual appearance and algebraic expression of transformed functions.
Stretching and compressing significantly affect both the visual appearance and algebraic expression of transformed functions. When a function is stretched vertically by multiplying it by a factor greater than 1 (e.g., $$af(x)$$ where $$a > 1$$), it becomes taller and retains its width; conversely, compressing it with a factor between 0 and 1 makes it shorter while keeping its width unchanged. Horizontally, compressing a function (e.g., $$f(bx)$$ where $$b > 1$$) makes it narrower, while stretching (e.g., $$f(bx)$$ where $$0 < b < 1$$) makes it wider. These transformations lead to altered dimensions but keep essential characteristics like zeros and continuity.
A transformation that flips the graph of a function over a specific axis, such as the x-axis or y-axis.
Stretching and Compressing: Transformations that alter the vertical or horizontal scale of a function's graph, either making it wider/narrower or taller/shorter.