USING TRANSFORMATIONS TO GRAPH ABSOLUTE VALUE FUNCTIONS

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.

Draw the graph of the following absolute value function using transformation.

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.

Moving the graph 2 units to the right.

Problem 2 :

y = |x + 4|

Solution :

Comparing the given function with y = |x|

h = -4 < 0

So, move the parent function 4 units to the left.

Problem 3 :

y = |x| + 3

Solution :

Comparing the given function with y = |x|

k = 3 > 0

So, move the parent function 3 units up.

Problem 4 :

y = |x| - 2

Solution :

Comparing the given function with y = |x|

k = -2 < 0

So, move the parent function 2 units down.

Problem 5 :

y = 2|x|

Solution :

Comparing the given function with y = |x|

Vertical stretch/compression :

a > 2

So, vertical stretch of 2 units. Then, it is narrower.

Problem 6 :

y = (-1/3)|x|

Solution :

Comparing the given function with y = |x|

Vertical stretch/compression :

a = 1/3 (0 < a < 1)

Reflection across x-axis.

So, vertical stretch of 2 units. Then, it is narrower.

Problem 7 :

y = 2|x - 2| - 3

Solution :

Comparing the given function with y = |x|

Problem 8 :

y = 3|x + 4| + 2

Solution :

Comparing the given function with y = |x|