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Translations

Discover and remember the pattern(s) by completing Help to Remember What to Learn

If you understand the pattern(s) the concept(s) in What to Learn should make sense.

What to Learn

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 (x-h)

Vertical

y y - k: Replace the y in the original function with (y-k)

(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 = (x-h)+k
or
y - k = (x - h)

2. y = (x-h)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|

  • not included in Math 30P

5.
or

6.
or

Source - Ron Blond

 

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 x-values

Function 5: x > h

Others: xR

Range - set of possible y-values

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 (x-h)

(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 = (x-h)+k
or
y - k = (x - h)

2. y = (x-h)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 |

  • not included in Math 30P

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 x-values

Function 5: x > h

Others: xR

Range - set of possible y-values

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 (y-k)

(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 = (x-h)+k
or
y - k = (x - h)

2. y = (x-h)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 |

  • not included in Math 30P

5.
or

6.
or

Linked Source - Ron Blond

Example 1: What is the new equation of the horizontal translation that could be applied to the graph of y = x2 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 = x2 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).

 
Math 30P   Sine/Cosine   Translations    Stretches   Reflections   Inverses   Reciprocals   Combinations   Prerequisites
 
 
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Started September, 1998. Copyright 2006, 2007