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2.6 FAMILIES OFFUNCTIONS
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FAMILIES OF FUNCTIONS There are sets of functions, called families that share certain characteristics. A parent function is the simplest form in a set of functions that form a family. Each function in the family is a transformation of the parent function
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TRANSLATIONS One type of transformation is a translation. A translation shifts the graph of the parent function horizontally, vertically, or both without changing shape or orientation.
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EXAMPLE How can you represent each translation of graphically?1. Shift the parent graph down 2 units2. Shift the parent graph left 1 unit
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EXAMPLE How can you represent each translation of graphically?1.Shift the parent graph right 3units and up 1 units2. Shift the parent graph left 2 units and down 3 units
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REFLECTIONS A reflection flips the graph over a line (such as the x – or y – axis) Each point on the graph of the reflected function is the same distance from the line of reflection as its corresponding point on the graph of the original function.
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When you reflect a graphREFLECTIONS in the y-axis, the x values change signs and the y- values stay the same. When you reflect a graph in the x-axis, the y-values change signs and the x- values stay the same.
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REFLECTING A FUNCTIONALGEBRAICALLY ( )Let f x = 3 x + 3 ( ) and g x be the reflection ( )in the x-axis. What is a function rule for g x ? g ( x) = − f ( x)
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REFLECTING A FUNCTIONALGEBRAICALLY ( ) ( )Let f x = 3 x + 3 and g x be the reflection in the ( )y-axis. What is a function rule for g x ? g ( x) = f ( −x)
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VERTICAL STRETCH ANDVERTICAL COMPRESSION A vertical stretch multiplies all y-values of a function by the same factor greater than 1. A vertical compression reduces all y-values of a function by the same factor between 0 and 1.Why do you think the value being multiplied isalways positive?
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EXAMPLEThe table represents the function f(x). Completethe table to find the vertical stretch and verticalcompression. Then graph the functions.
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EXAMPLE: COMBININGTRANSFORMATIONSThe graph of g(x) is the graph of f(x) = 4xcompressed vertically by the factor ½ and thenreflected in the y-axis. What is the function rulefor g(x)?
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EXAMPLE: COMBININGTRANSFORMATIONSThe graph of g(x) is the graph of f(x) = x stretchedvertically by the factor 2 and then translateddown 3 units. What is the function rule for g(x)?
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