2-6 Families of Functions Hubarth Algebra 2

A translation shifts a graph horizontally, vertically or both A translation shifts a graph horizontally, vertically or both. It’s a graph of the same shape and size but possibly in a different position. Ex 1. Vertical Translation a. Describe the translation 𝑦= 𝑥 −3 and draw its graph. b. Write an equation to translate 𝑦= 𝑥 up 1 2 units up. This is a translation of 𝑦= 𝑥 by 3 units down. 𝑦= 𝑥 + 1 2

Ex 2. Horizontal Translation a. The blue graph at the right is a translation of 𝑦= 𝑥 . Write the equation for the graph. b. Describe the translation 𝑦= 𝑥+3 and draw its graph. 𝑦= 𝑥−5 𝑦= 𝑥+3 is a translation of 𝑦= 𝑥 by 3 units to the left

Summary The Family of Absolute Value Functions Vertical Translation Parent function: 𝑦= 𝑥 𝑦=𝑓(𝑥) Translation up k units, k>0: 𝑦= 𝑥 +𝑘 𝑦=𝑓 𝑥 +𝑘 Translation down k units, k>0: 𝑦= 𝑥 −𝑘 𝑦=𝑓 𝑥 −𝑘 Horizontal Translation Translation right h units, h>0: 𝑦= 𝑥−ℎ 𝑦=𝑓(𝑥−ℎ) Translation left h units, h>0: 𝑦= 𝑥+ℎ 𝑦=𝑓(𝑥+ℎ) Combined Translation (right h units, up k units) 𝑦= 𝑥−ℎ +𝑘 𝑦=𝑓 𝑥−ℎ +𝑘

A vertical stretch multiplies all y-values by the same factor greater than 1, therefore stretching the graph vertically. A vertical shrink reduces the y-values by a factor between O and 1, thereby compressing the graph vertically. Ex 3. Graphing 𝒚=𝒂 𝒙 a. Describe and then draw the graph of 𝑦=2 𝑥 . 𝑦=2 𝑥 is a vertical stretch of 𝑦= 𝑥 by a factor of 2. each y-value of 𝑦=2 𝑥 is twice the corresponding y-values of 𝑦= 𝑥 . *NOTE* (2, 2) on 𝑦= 𝑥 , whereas (2,4) lies on 𝑦=2 𝑥 . b. Write an equation for a vertical shrink of 𝑦= 𝑥 by a factor of 1 2 𝑦= 1 2 𝑥

Ex 4. Graphing 𝒚=−𝒂 𝒙 Which equation describes the graph? 𝑎. 𝑦= 1 2 𝑥 𝑦= 1 2 𝑥 𝑦=− 1 2 𝑥 𝑦=− 1 2 𝑥 𝑐. 𝑦=− 1 2 𝑥

Summary Families of Functions: Absolute Values Vertical Stretch or Shrink, and Reflections in x-axis Parent function: 𝑦= 𝑥 y=𝑓(𝑥) Reflection across x-axis: 𝑦=− 𝑥 𝑦=−𝑓 𝑥 Stretch (a > 1) Shrink (0 < a < 1) Reflection across x-axis: 𝑦=−𝑎 𝑥 𝑦=−𝑎𝑓 𝑥 Combined Transformation 𝑦=𝑎 𝑥−ℎ +𝑘 𝑦=𝑎𝑓 𝑥−ℎ +𝑘 by factor a: 𝑦=𝑎 𝑥 𝑦=𝑎𝑓(𝑥)

Practice 1. Write the equation for the graph 𝑦= 𝑥+3 2. Write an equation for the vertical stretch of 𝑦= 𝑥 by a factor of 3 𝑦=3 𝑥 3. A function is a vertical stretch of 𝑦= 𝑥 by a factor of 5. Write an equation for the reflection of the function across the x-axis. 𝑦=−5 𝑥