Lesson 2-6 Families of Functions

A family of functions is made up of functions with certain common characteristics. Parent function – simplest function with these characteristics. The equations of the functions in a family resemble each other

Our parent function is y = |x| A translation shifts a graph horizontally, vertically, or both. Consider the functions y = |x| y = |x| + 2 y = |x| – 2 y = |x + 4| y = |x – 4|

Horizontal Translation Vertical Translation Translation up k units y = |x| + k Translation down k units y = |x| – k Horizontal Translation Translation right h units y = |x – h| Translation left h units y = |x + h| Combined Translation Both a horizontal and vertical shift. y = |x – 2| + 3

A vertical stretch multiplies all values by the same factor greater than 1. A vertical shrink reduces y-values by a factor between 0 and 1. Consider the functions y = |x| y = 2|x| y = 4|x| y = ½|x| y = ¼|x|

A reflection in the x-axis changes y values to their opposites. y = |x| y = 2|x| y = -2|x|

Vertical Stretch Vertical Shrink Reflection Stretch by a factor of a y = a|x| Note a must be greater than 1 Vertical Shrink Shrink by a factor of a y = a|x| Note a must be between 0 -1 Reflection Reflect on the x-axis y = -a|x|

Summary of how to graph absolute values. Find the vertex Use the number out front as the slope Get points on both the left and right side Draw the graph Note this method only works when x is alone inside the absolute value symbols