1. 2-6 Families of Functions
Hubarth
Algebra 2

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 units up.
This is a translation of 𝑦= 𝑥 by 3 units down.
𝑦= 𝑥 + 1 2

3. 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

4. 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

5. 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

6. 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

7. 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)

8. 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