5.2.4 Transforming Linear Functions Key Objectives: • Translate, rotate, and reflect linear functions. • Describe transformations of linear functions. • Perform multiple transformations of linear functions. • Write equations of transformed functions. Key Terms
• A transformation is a change in position or size of a figure. • A rotation is a transformation of a figure about a point. • A reflection is a transformation across a line that produces a mirror image. • A translation is a transformation that shifts every point of a figure the same distance and direction along a straight line. A family of functions is a set of functions whose graphs have basic characteristics in common. For example, all linear functions form a family because all of their graphs are the same basic shape (a line). The most basic function in each family of functions is called the parent function. For linear functions, the parent function is y = x , or in function notation f ( x ) = x . The graphs of all other linear functions are transformations of the graph of the linear parent function f ( x ) = x . A transformation of a graph is a change in its position. So, the position of the graph of any linear function has been changed in some way as compared to the graph of f ( x ) = x . Three types of transformations of linear functions (translations, rotations, and reflections) are discussed in the examples below. Example 1 Translating Linear Functions A translation is a type of transformation that moves every point on the graph the same distance in the same direction. A translation is often thought of as a slide of a graph. Translations of functions can be horizontal or vertical. A horizontal translation moves a graph left or right. A vertical translation moves a graph up or down. Changing the y -intercept b in the function f ( x ) = mx + b translates the line vertically. If b increases, the line is translated up. If b decreases, the line is translated down. Moreover, any linear function where b is positive is a translation of f ( x ) = x up b units, and any linear function where b is negative is a translation of f ( x ) = x down b units. Note that the slope of a line is not changed by a translation.
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