Examples
f(x)=|x|f(x)=|x| , g(x)=|x|-5g(x)=|x|−5
Step 1
The transformation from the first equation to the second one can be found by finding aa, hh, and kk for each equation.
y=a|x-h|+ky=a|x−h|+k
Step 2
Factor a 11 out of the absolute value to make the coefficient of xx equal to 11.
y=|x|y=|x|
Step 3
Factor a 11 out of the absolute value to make the coefficient of xx equal to 11.
y=|x|-5y=|x|−5
Step 4
Find aa, hh, and kk for y=|x|-5y=|x|−5.
a=1a=1
h=0h=0
k=-5k=−5
Step 5
The horizontal shift depends on the value of hh. When h>0h>0, the horizontal shift is described as:
g(x)=f(x+h)g(x)=f(x+h) - The graph is shifted to the left hh units.
g(x)=f(x-h)g(x)=f(x−h) - The graph is shifted to the right hh units.
Horizontal Shift: None
Step 6
The vertical shift depends on the value of kk. When k>0k>0, the vertical shift is described as:
g(x)=f(x)+kg(x)=f(x)+k - The graph is shifted up kk units.
g(x)=f(x)-kg(x)=f(x)−k - The graph is shifted down kk units.
Vertical Shift: Down 55 Units
Step 7
The sign of aa describes the reflection across the x-axis. -a−a means the graph is reflected across the x-axis.
Reflection about the x-axis: None
Step 8
The value of aa describes the vertical stretch or compression of the graph.
a>1a>1 is a vertical stretch (makes it narrower)
0<a<10<a<1 is a vertical compression (makes it wider)
Vertical Compression or Stretch: None
Step 9
To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.
Parent Function: f(x)=|x|f(x)=|x|
Horizontal Shift: None
Vertical Shift: Down 55 Units
Reflection about the x-axis: None
Vertical Compression or Stretch: None
Step 10
