2-6: Families of Functions

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Presentation transcript:

2-6: Families of Functions Essential Question: What ways can the graph of the parent function y = |x| be transformed?

2-6: Families of Functions A parent function is a function with a certain shape that has the simplest rule for that shape. Other equations will resemble the parent function in both formula as well as shape. The parent function for an absolute value graph is y = |x| A translation is a shift of the parent function either horizontally (left/right), vertically (up/down) or both. It results in a graph with the same size and shape, but in a different location The translations we will cover here today apply to both y = |x| as well as y = -|x|

2-6: Families of Functions Vertical translations For a positive number k, y = |x| + k, is a vertical translation If k is a positive number, shift the graph up k units If k is a negative number, shift the graph down k units Vertical translations work as expected parent function: y = |x| y = |x| - 3 vertical shift down 3 units y = |x| + 2 vertical shift up 2 units

2-6: Families of Functions Your Turn: Describe the translation: y = |x| + 1 y = |x| - 2 Write an equation for the translation of: y = |x| up 8 units y = |x| down ½ unit y = |x| shifted up 1 unit y = |x| shifted down 2 units y = |x| + 8 y = |x| - ½

2-6: Families of Functions Horizontal translations For a positive number k, y = |x + h|, is a horizontal translation (horizontal change is on the inside) If h is a positive number, shift the graph left h units If h is a negative number, shift the graph right h units Horizontal translations work opposite of expected “HI HO” parent function: y = |x| y = |x + 3| horizontal shift left 3 units y = |x – 2| horizontal shift right 2 units

2-6: Families of Functions Your Turn: Describe the translation: y = |x + 3| y = |x – 1| Write an equation for the translation of: y = |x| right 2 units y = |x| left 4 units y = |x| shifted left 3 units y = |x| shifted right 1 unit y = |x – 2| y = |x + 4|

2-6: Families of Functions Putting it all together Describe the translation of f(x) = |x| y = |x + 2| + 4 y = |x – 6| + 5 Horizontal shift 2 units left Vertical shift 4 units up Horizontal shift 6 units right Vertical shift 5 units up

2-6: Families of Functions Assignment Page 97 – 98 Problems 1 – 14 & 29 – 34 All problems Instead of graphing (1 – 4, 8 – 11, 29 – 34), simply list the transformations that occur (e.g. “vertical shift down 2 units”) like what we did earlier