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UNIT 2 : FUNCTIONS
and TRANSFORMATIONS
LESSON 3a: TRANSFORMATIONS OF QUADRATIC FUNCTIONS
Vertical Translations /shifts of Quadratic
Functions:
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).
Horizontal Translations /shifts of Quadratic
Functions:
Note that
all three graphs are congruent. The
graph of y = (x -2 )2
(green) has been shifted right 2 units relative to the
graph of y = x2 (blue). This transformation
(change) is called a horizontal translation (shift) up 3 units.
he
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 + 2, y )
Since the
translation is right 2, we add 2 to the x-coordinate of each point on the
original function.
The graph
of y = (x + 3)2 (red) has been
shifted left 3 units relative to the graph of y = x2.
This
transformation (change) is called a horizontal translation (shift) left 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 - 3, y)
Example
1: 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 2:
(x,
y) ---------------------------à(x ,
y – 3)
(-2, 4) ---------------------à (-2, 4-3) = (-2,1)
(-1, 1)
---------------------à (-1 , 1-3)
= (-1,-2)
(0,0) -----------------------à (0, 0-3) = (0,-3)
(1, 1)
----------------------à (1, 1-3) = (1,-2)
(2, 4)
----------------------à (2 , 4-3)
= (2,1) yielding the
graph at left with vertex at (0, -3)
(x,
y) ---------------------------à(x - 3
, y - 2)
(-2, 4)
---------------------à (-2-3, 4-2) = (-5,2)
(-1, 1)
---------------------à (-1 -3, 1-2) = (-4,-1)
(0,0) -----------------------à (0-3, 0-2) = (-3,-2)
(1, 1)
----------------------à (1-3, 1-2) = (-2,-1)
(2, 4)
----------------------à (2 -3, 4-2) = (-1,2) yielding the graph at left with
vertex at (-3, -2)
Stretches and Compressions of
Quadratic Functions:
Example
3: Given the graph of f(x) = x2 as shown,
draw the graphs of y = 3f(x), and y = ½ f(x)
Solution:
The
required graph of f(x) = 3x2 will be a vertical stretch factor 3 and may be represented
in mapping form:
(x, y)
---------------------------(x, 3y)
(Now
take key points on the graph of y = x2 -- {(-2, 4), (-1, 1), (0, 0), (1,
1), (2, 4)} and multiply the y-coordinates by 3.
(x, y) ----------------------------à(x, 3y)
(-2,
4) ---------------------------à(-2, 12)
(-1,
1) ---------------------------à(-1, 3)
(0,
0) ----------------------------à(0, 0)
(1,
1) ----------------------------à(1, 3)
(2,
4) ----------------------------à(2, 12) [see red graph y = 3x2 at
left]
The
required graph of y = ½ x2 will be a vertical compression
factor ½ and may be represented in
mapping form:
(x, y)
---------------------------(x, ½ y)
(Now
take key points on the graph of y = x2 -- {(-2, 4), (-1, 1), (0, 0), (1,
1), (2, 4)} and multiply the y-coordinates by ½ or 0.5
(x, y) -----------------------------à(x, 3y)
(-2, 4) ---------------------------à(-2, 2)
(-1, 1) ---------------------------à(-1, 0.5)
(0, 0) ----------------------------à(0, 0)
(1, 1) ----------------------------à(1, 0.5)
(2, 4) ----------------------------à(2, 2) [see green graph y = ½ x2
above left]
.
In summary, if
k is an integer: The graph of kf(x) will
be a vertical
stretch of factor k and the graph of 1/k
f(x) will be a vertical compression of factor 1/k
COMBINATIONS OF TRANSFORMATIONS
