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LESSON 3: TRANSLATIONS
Vertical Translations /shifts:
Example 1:
Note that all three graphs are congruent. The graph of y = x2 + 3 (green) has been shifted up 3
units relative to the graph of y =
x2 (blue). This transformation (change) is called a vertical
translation (shift) up 3 units. The transformation may be depicted in mapping form.
Each point (x, y) on the base curve y = x2 has been transformed as follows:
(x, y) -------------------------------à (x, y + 3)
Since the translation is up 3, we add 3 to the y-coordinate of each point on the original function.
The graph of y = x2 – 2(red) has been shifted down 2 units relative to the graph of y = x2.
This transformation (change) is called a vertical translation (shift) down 2 units.
The transformation may be depicted in mapping form. Each point (x, y) on the base curve y = x2
has been transformed as follows:
(x, y) -------------------------------à (x, y - 2)
Since the translation is down 2, we simply subtract 2 from the y-coordinate of each point on the original function.
In general, the graph of y = x2 + k is a vertical translation of “k” units up or down relative to the base curve y = x2.
More generally, for any function f(x), the graph of f(x) + k is a vertical translation of “k” units (up or down) relative to f(x).
In mapping form: (x, y) ----------------------(x, y – 1)
To get the graph, recall the key
points {(0,0), (1,1), (4,2), (9, 3), (16,4)}.
Now use the mapping as a formula and apply it to each of the
key points as follows.
(x, y)
----------------------------à(x, y – 1)
(0, 0) --------------------------à(0, 0 – 1) = (0, -1)
(1, 1) --------------------------à(1, 1 – 1) = (1, 0)
(4, 2) --------------------------à(4, 2 – 1) = (4, 1)
(9, 3) --------------------------à(9, 3 – 1) = (9, 2)
(16, 4) -------------------------à(16, 4 – 1) = (16, 3)
In mapping form: (x, y) --------------------à (x, y + 2)
To get the graph, recall the key
points for y = 1/x , namely{(-2,-1/2), (-1,-1), (-1/2,-2), center (0,
0),(1/2,2),(1,1),(2,1/2)}.
Now use the mapping as a formula
and apply it to each of the key points as follows.
Note that (0, 0) is not actually a
point on the curve, but acts as a center for the graph. The x-axis (y=0) and the y-axis (x=0)
act as asymptotes to the graph –
the curve gets closer and closer to these lines.
(x, y)
-------------------------------à(x, y + 2)
(-2, -½) --------------------------à(-2, -½ + 2) = (-2, 1 ½) or (-2, 1.5)
(-1, -1)
----------------------------à(-1, -1 + 2) = (-1, 1)
(-1/2, -2)
--------------------------à(-1/2, -2 + 2) = (-1/2,0)
(0, 0)
------------------------------à(0, 0 + 2) = (0, 2)
center
(1/2 , 2)
---------------------------à(1/2, 2 + 2) = (1/2, 4)
(1, 1)
------------------------------à(1, 1 + 2) = (1, 3)
(2, ½)
-----------------------------à(2, ½ + 2) = (2, 2 ½)
Domain: Consider the graph in the horizontal (x) direction. The graph extends to infinity
to the left and to the right. However there is a break in the graph as it does not cross the y-axis.
The domain will be all real numbers except x = 0 (y-axis). (vertical
asymptote).
Range: Consider the graph in the y-direction. The graph extends to infinity both up and down.
Similarly there is a break in the graph where y = 2 (horizontal asymptote).
Horizontal Translations /Shifts:
If x = -2, y = 0 giving the ordered pair (-2, 0)
If x = -1, y = 1 giving the ordered pair (-1,1)
If x = 2, y = 2 giving the ordered pair (2,2)
………….etc. yielding the table
and graph below.
x |
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-2 |
0 |
-1 |
1 |
2 |
|
7 |
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14 |
|
.
(0, 0) --------------------------à(0-2, 0) = (-2, 0)
(1, 1) --------------------------à(1 - 2, 1 ) = (-1, 1)
(4, 2) --------------------------à(4 -2, 2 ) = (2, 2)
(9, 3) --------------------------à(9 - 2, 3) = (7, 2)
(16, 4) -------------------------à(16 - 2, 4 ) = (14, 4)
.
Example 6: Sketch the graph of y = (x + 1)2. Refer to graph of y = x2 above. This will be a translation left 1 unit relative to this graph. In mapping form:
(x, y) ------------------------à (x – 1, y) and using the basic points for y = x2 {(-2, 4), (-1, 1), (0, 0), (1, 1), (2,4)}, we obtain
(-2, 4) ---------------------à (-2 – 1, 4) = (-3,4)
(-1, 1) ---------------------à (-1 – 1, 1) = (-2,1)
(0,0) ---------------------à (0– 1, 0) = (-1,0)
(1, 1) ---------------------à (1 – 1, 1) = (0,1)
(2, 4) ---------------------à (2 – 1, 4) = (1,4) yielding the graph below with vertex at (-1, 0)
Example 7 – Combinations: Given the function y = f(x) below [blue], sketch the graph of f(x – 2) – 1.
The required graph will be a shift
right 2 and down 1 and may be represented in mapping form:
(x, y)
---------------------------(x + 2, y – 1)
Now take key points on the graph
of y = f(x) -- {(-5, 1), (-3, 1), (-1,
3),(0, 2), (2, 0)}
and move each one right 2 and down
1 using the mapping.
(x, y)
---------------------------(x + 2, y – 1)
(-5, 1)
---------------------------(-5 + 2, 1 – 1) = (-3, 0)
(-3, 1) -------------------------à(-3 + 2, 1 – 1) = (-1, 0)
(-1, 3) -------------------------à(-1 + 2, 3 – 1) = (1,
2)
(2, 0) --------------------------à (4, -1) giving the graph below [red]