The function h(x)=f(x-a) transforms f(x) by shifting it to the right a units.
C
c
The function j(x)=- f(x) transforms f(x) by reflecting it over the x-axis.
D
d
The function k(x)=a* f(x) transforms f(x) by vertically stretching it by a factor of a.
A
a
A'(- 1/2,- 2), B'(1,- 5), C'(- 4, - 7)
B
b
A'(21/2,3), B'(4,0), C'(- 1, - 2)
C
c
A'(- 1/2,- 3), B'(1,0), C'(- 4, 2)
D
d
A'(- 1/2,12), B'(1,0), C'(- 4,- 8)
Practice makes perfect
a
Let's look at all of the possible transformations so that we can more clearly identify what is happening to our function.
Transformations of f(x)
Vertical Translations
Translation up k units, k>0
y=f(x)+ k
Translation down k units, k>0
y=f(x)- k
Horizontal Translations
Translation right h units, h>0
y=f(x- h)
Translation left h units, h>0
y=f(x+ h)
Vertical Stretch or Compression
Vertical stretch, a>1
y= af(x)
Vertical compression, 0< a< 1
y= af(x)
Horizontal Stretch or Compression
Horizontal stretch, 0< b<1
y=f( bx)
Horizontal compression, b>1
y=f( bx)
Reflections
In the x-axis
y=- f(x)
In the y-axis
y=f(- x)
Now let's look at the given function.
g(x)=f(x)- 5
We can see that this is a vertical translation 5 units down of f. To determine the points of g, we can decrease the outputs, or y-coordinates, of the points of f by 5. The x-coordinates will remain the same, because no horizontal shift takes place. The following table of values gives the transformed points.
h(x)=f(x - 3)
Looking at the table in Part A, we can see that this is a horizontally translation of f 3 units to the right. To determine the points of h we can increase the x-coordinates of the points of f by 3. The y-coordinates will remain the same, because no vertical shift takes place. The following table gives the transformed points.
Point
f(x)
f(x-3)
h(x)
A
(- 1/2,3)
(- 1/2+ 3,3)
(2 12,3)
B
(1,0)
(1+ 3,0)
(4,0)
C
(- 4, - 2)
(- 4+ 3, - 2)
(- 1, - 2)
Let's graph these points on the coordinate plane.
c
Looking at the table from Part A we know that the function j(x)=- f(x) transforms f(x) by reflecting it over the x-axis. To determine the points of j we can multiply the y-coordinates of the points of f by - 1. The following table gives the transformed points.
Point
f(x)
- 1 f(x)
j(x)
A
(- 1/2,3)
(- 1/2,3* ( - 1))
(- 1/2,- 3)
B
(1,0)
(1,0* ( - 1))
(1,0)
C
(- 4, - 2)
(- 4,- 2* ( - 1))
(- 4, 2)
Let's plot these points in the coordinate plane.
d
Let's look at the given function.
k(x)= 4f(x)Looking at the table from Part A, we can see that this is a vertical stretch of f by a factor of 4. To determine the points of k we can multiply the y-coordinates of the points of f by 4. The following table gives the transformed points.
