DESCRIBE THE TRANSFORMATION OF THE ABSOLUTE VALUE FUNCTION

Translation

A translation is a  transformation  that shifts a graph  horizontally or  vertically, but  doesn’t change  the overall shape  or orientation.

• If h > 0, then move the graph horizontally towards the right.
• If h < 0, then move the graph horizontally towards the left.
• If k > 0, then move the graph vertically up.
• If k < 0, then move the graph vertically down.

Example problems on horizontal and vertical translation of absolute value functions

Horizontal and Vertical Shrink or Stretches

The function which is in the form y = |ax - h| + k

Here a > 1 or 0 < a < 1

• If a > 1, the graph should be narrower or horizontal shrink
• If 0 < a < 1, the graph should be wider or horizontal stretch.

The function which is in the form y = a|x - h| + k

Here a > 1 or 0 < a < 1

• If a > 1, the graph should stretch vertically or narrower.
• If 0 < a < 1, the graph should have vertical shrink or wider.

Describe the 

Problem 1 :

y = |x - 2|

Solution :

Comparing the given function with the parent function y = |x|

h = 2, so move the graph 2 units to the right.

Problem 2 :

y = |x|+3

Solution :

Comparing the given function with the parent function y = |x|

h = 3, so move the graph 3 units up.

Problem 3 :

y = 2|x + 3|

Solution :

Comparing the given function with the parent function y = |x|

h = -3, so move the graph 3 units left.

a = 2 > 1 vertical stretch 2 units, so it is narrower.

Problem 4 :

y = 3|x|

Solution :

Comparing the given function with the parent function y = |x|

a = 3 > 1 vertical stretch 3 units, so it is narrower.

Problem 5 :

y = -2|x + 3| - 1

Solution :

Comparing the given function with the parent function y = |x|

• a = 2 > 1 vertical stretch 2 units, it is narrower.
• a = -2, reflection across x-axis.
• h = -3, horizontal translation of 3 units left.
• k = 1, vertical translation of 1 unit up.

Problem 6 :

y = 2|x + 8|

Solution :

Comparing the given function with the parent function y = |x|

• a = 2 > 1 vertical stretch 2 units, it is narrower.
• h = -8, horizontal translation of 8 units left.
  

Problem 7 :

y = -|x - 6| + 2

Solution :

Comparing the given function with the parent function y = |x|

• a = -1, reflection across x-axis. 
• h = 6, horizontal translation of 6 units right.
• k = 2, vertical translation of 2 unit up.

Problem 8 :

y = (1/2) |x + 2| - 1

Solution :

Comparing the given function with the parent function y = |x|

• a = 1/2, vertical shrink of 1/2 units. 
• h = -2, horizontal translation of 2 units left.
• k = -1, vertical translation of 2 unit down.