1. 2-6 Families of Functions M11.D.2.1.2: Identify or graph functions, linear equations, or linear inequalities on a coordinate plane

2. Objectives Translations Stretches, Shrinks, and Reflections

3. Vocabulary A family of functions is made up of functions with certain common characteristics. A parent function is the simplest function with these characteristics. The equations of the functions in the family resemble each other. So do the graphs. A translation shifts the graph horizontally, vertically, or both. It results in a graph of the same shape and size, but possibly a different position.

4. Vertical Translation y = |x| – 2 is a translation of y = |x| by 2 units downward. Each y-value for y = |x| – 2 is 2 less than the corresponding y-value for y = |x|. a.Describe the translation y = |x| – 2 and draw its graph by translating the parent function. b. Write an equation for the translation of y = |x| up 8 units. An equation that translates y = |x| up 8 units is y = |x| + 8. Vertical Translation

5. Horizontal Translation a. Describe the translation y = |x + 4| and draw its graph by translating the parent function. y = |x + 4| is a translation of y = |x| by 4 units left. Each x-value for y = |x + 4| is 4 less than the corresponding x-value for y = |x|. b. Write an equation for the translation of y = |x| right 2 units. An equation that translates y = |x| right 2 units is y = |x – 2|. Horizontal Translation

6. Notes from the Book Take a few minutes to write down the information inside the orange box on page 95. These notes may come in handy during your test.

7. Describe a possible translation of Figures M and N in the design shown below. Translate Figure M 2 units down, For Figure N, there are two possible translations: 4 units down, or else 1 unit right and 2 units down. Real World Example

8. Vocabulary A vertical stretch multiplies all y-values by the same factor greater than 1, thereby stretching the graph vertically. A vertical shrink reduces the y-values by a factor between 0 and 1, thereby compressing the graph vertically More formally, for the parent function y = |x| and a number a, a > 1, y = a|x| is a vertical stretch 0 < a < 1, y = a|x| is a vertical shrink

9. Graphing y = a|x| a. Describe and then draw the graph of y = |x|. b. Write an equation for a vertical stretch of y = |x| by a factor of 6. A vertical stretch of y = |x| is y = 6|x|. 1414 y = |x| is a vertical shrink of y = |x| by a factor of. Each y-value for y = |x| is one-fourth the corresponding y-value for y = |x|. 1414 1414 1414 Graphing y = a|x|

10. Vocabulary A reflection in the x-axis changes y-values to their opposites. When you change the y-values of the graph to their opposites, the graph to their opposite, the graph reflects across the x-axis.

11. Write the equation for the graph. The parent function is y = |x|. The graph shows a vertical stretch of 3 units and a reflection over the x-axis. The equation of the graph is y = –3|x|. Graphing y = -a|x|

12. Notes from the Book Take a few minutes to write down the information inside the orange box on page 97. These notes may come in handy during your test.

13. Homework Pg 97 & 98 # 1,5,8,12,15,17