
Help to Remember What to Learn

Standard Form
of Translations of Functions
Parameters:
transformed function 
y = f(x  h) + k

h 
horizontal translation parameter
 Function 5: changes domain, but not
range
 Remainder of functions: does not change domain or range.

k 
vertical translation parameter
 Function 2, 4, 5: changes range, but not
domain
 Remainder of functions: does not change domain or range.

(h, k)

Location of red data point on graph
 is at the origin when (h, k)
= (0, 0). At this point (h, k)
is the center of the graph.

Description of Translation/Shift 
Horizontal
 If h > 0, then (x  h):
translation h units
right.
 else if h = 0, then (x  0)
= (x): no horizontal translation.
 else if f h < 0, then (x  h):
translation h units left.
Vertical
 If k > 0, then (x  k):
translation k units up
 else if k = 0, then (x)  0 =
(x): no vertical translation.
 else if f k < 0, then (x  k):
translation k units down.

Algebraic Description of Translation/Shift 
Horizontal
x x
 h:
Replace the x in the original function with (xh)
Vertical
y y  k: Replace
the y in the original function with (yk) 
(x, y) (x+h,
y+k)
blue data point red
data point

mapping notation to find location of data points
after a horizontal/vertical translation. Notice the green rectangle
connecting the data points is h units wide
and k units high.

 translation: the transformation of a geometric figure in which
every point is moved in the same distance in
the same
direction.

Interactive Activity
 Click on the applet to activate
 Click one of the 6 function icons.
 Move the parameter sliders for h and k.
 Study the effects of each parameter with respect to the original
equation (grey graph). Determine the parameter that effects:
 horizontal translation (shift left/right)
 vertical translation (shift up/down)
 Drag the blue data point from the far left to the far right. Observe
the movement of the red data point to see how the transformation
can be used to map the location of corresponding data points on the
transformed function. The location(ordered pairs) of the blue and
red data points is given at the bottom of the screen.
 Click the red or blue data point to update the red
data point if you use the up/down arrow to move one of the sliders.
 Select SET FUNTION to explore the effect of the
h and k parameters on other functions.
1. y = (xh)+k or y  k = (x  h)

2. y = (xh)^{2}+k or y  k = (x  h)

3. y = (x h) ^{3}+ k
or
y  k = (x  h) ^{3}


4. y = x  h+k
or
y  k = x  h

5. or

6. or


Horizontal
Translation (Slide)
In both functions, h is the horizontal translation
parameter.
transformed function 
y = f(x  h) or x x
 h 
horizontal translation parameter 
h

Domain  set of possible xvalues 
Function 5: x > h
Others: xR 
Range  set of possible yvalues

Functions 2, 4, 5: y > k
Others: yR

(h, k)

Location of red data point on graph

Description of Horizontal Translation/Shift 
If h > 0, then (x  h):
translation h units right.
 When h = 3, (x  3): translation 3 units
right
else if h = 0, then (x  0)
= (x): no horizontal translation.
else if f h < 0, then (x  h):
translation h units left.
 When h = 3, (x  3) = (h + 3): translation
3 units left

Algebraic Description of Horizontal Translation/Shift 
x x
 h: Replace the x in the original function with
(xh)

(x, y) (x+h,
y +k)
blue data point red
data point

mapping notation to find location of data points after a horizontal/vertical
translation. Notice the green rectangle
connecting the data points is h units wide and k units
high.


Interactive Activity
 Click on the applet to activate
 Click one of the 6 function icons.
 Move the parameter sliders for h. Note
the original equation remains and a second graph is drawn according
to the parameters selected.
 Drag the blue data point from the far left to the far right. Observe
the movement of the red data point to see how the transformation can
be used to map the location of corresponding data points on the transformed
function. The location(ordered pair) of the blue and red data point
is given at the bottom of the screen.
 Select SET FUNTION to explore the effect of the h parameter
on other functions.
1. y = (xh)+k
or
y  k = (x  h)

2. y = (xh)^{2}+k
or
y  k = (x  h)

3. y = (x h) ^{3}+ k
or
y  k = (x  h) ^{3}


4. y = x  h+k
or
y  k =  x  h 

5.
or

6.
or

Linked
Source  Ron Blond

Vertical
Translation (Slide)
In both functions, k is the vertical translation parameter.
transformed function 
y = f(x) + k or y y
 k 
vertical displacement/translation parameter 
k

Domain  set of possible xvalues 
Function 5: x > h
Others: xR 
Range  set of possible yvalues

Functions 2, 4, 5: y > k
Others: yR

(h, k)

Location of red data point on graph

Description of Vertical Translation/Shift 
If k > 0, then (x) + k:
translation k units up
 When k = 3, (x) + 3: translation 3 units
up
else if k = 0, then (x)  0 =
(x): no vertical translation.
else if f k < 0, then (x) + k:
translation k units down.
 When k = 3, (x)  3: translation
3 units down

Algebraic Description of Vertical Translation/Shift 
y y  k: Replace
the y in the original function with (yk)

(x, y) (x
+ h, y + k)
blue data point red
data point

mapping notation to find location of data points after a horizontal/vertical
transformation. Notice the green rectangle
connecting the data points is h units wide and k units
high.

Battle of the Opposites in ( x  h )
 the operation
between x and h is subtraction thus (x  3) is a shift to the right,
while (x + 3) is
a shift to the left.

Interactive Activity
 Click on the applet to activate
 Click one of the 6 function icons.
 Move the parameter sliders for k. Note the original
equation remains and a second graph is drawn according to the parameters
selected.
 Drag the blue data point from the far left to the far right. Observe
the movement of the red data point to see how the transformation can
be used to map the location of corresponding data points on the transformed
function. The location(ordered pair) of the blue and red data point
is given at the bottom of the screen.
 Select SET FUNTION to explore the effect of the k parameter
on other functions.
1. y = (xh)+k
or
y  k = (x  h)

2. y = (xh)^{2}+k
or
y  k = (x  h)

3. y = (x h) ^{3}+ k
or
y  k = (x  h) ^{3}


4. y = x  h+k
or
y  k =  x  h 

5.
or

6.
or


Example 1: What
is the new equation of the horizontal translation that could be applied
to the graph of y = x^{2} so
that the translation passes through the point (3, 0). Describe
the horizontal translation that could be applied to the original graph.
Solution:
If the graph of y = (x  h)^{2} passes through the point (3,
0), then
0 = (3  h)^{2}
0 = 3  h
h = 3
The new equation of the vertical translation is:
y = (x  3)^{2}
A horizontal translation of 3 units right would result in
a translation image that passes through (3, 0). 
Example 2:
What is the new equation of the vertical translation that could
be applied to the graph of y = x^{2} so that the translation
passes through the point (0, 3). Describe the horizontal translation
that could
be applied to the original graph.
Solution:
If the graph of y = (x)^{2} + k passes through the point (3,
0), then
0 = (3)^{2} + k
0 = 9 + k
k = 9
The new equation of the vertical translation is:
y = (x)^{2}  9
A vertical translation of 9 units down would result in a translation
image that passes through (3, 0). 