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Functions Defined

Functions You Should Know

Translations

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Combining Functions

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 UNIT 2 : FUNCTIONS

 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) ------------------------------- (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) ------------------------------- (x, y - 2)

 

 

Since the translation is down 2, we simply subtract 2 from the y-coordinate of each point on the original function.

Text Box: 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.
 

 

 

 

 

 

 


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) ----------------------------(x, y 1)

(0, 0) --------------------------(0, 0 1) = (0, -1)

(1, 1) --------------------------(1, 1 1) = (1, 0)

(4, 2) --------------------------(4, 2 1) = (4, 1)

(9, 3) --------------------------(9, 3 1) = (9, 2)

(16, 4) -------------------------(16, 4 1) = (16, 3)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

In mapping form: (x, y) -------------------- (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) -------------------------------(x, y + 2)

(-2, -) --------------------------(-2, - + 2) = (-2, 1 ) or (-2, 1.5)

(-1, -1) ----------------------------(-1, -1 + 2) = (-1, 1)

(-1/2, -2) --------------------------(-1/2, -2 + 2) = (-1/2,0)

(0, 0) ------------------------------(0, 0 + 2) = (0, 2) center

(1/2 , 2) ---------------------------(1/2, 2 + 2) = (1/2, 4)

(1, 1) ------------------------------(1, 1 + 2) = (1, 3)

(2, ) -----------------------------(2, + 2) = (2, 2 )

 

 

 

 

 

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

-2

0

-1

1

2

 

7

 

14

 

 

 

 

 

 

 

 

 

 

 

 

 

.

 

(0, 0) --------------------------(0-2, 0) = (-2, 0)

(1, 1) --------------------------(1 - 2, 1 ) = (-1, 1)

(4, 2) --------------------------(4 -2, 2 ) = (2, 2)

(9, 3) --------------------------(9 - 2, 3) = (7, 2)

(16, 4) -------------------------(16 - 2, 4 ) = (14, 4)

 

 

.

 

 

Text Box: In general the transformation y = f(x - h)  produces a horizontal translation h units in the opposite direction of the sign before h.  In mapping form

(x, y) -------------------------- (x + h, y)
 

 

 

 

 

 

 


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) ------------------------ (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) --------------------- (-2 1, 4) = (-3,4)

(-1, 1) --------------------- (-1 1, 1) = (-2,1)

(0,0) --------------------- (0 1, 0) = (-1,0)

(1, 1) --------------------- (1 1, 1) = (0,1)

(2, 4) --------------------- (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) -------------------------(-3 + 2, 1 1) = (-1, 0)

(-1, 3) -------------------------(-1 + 2, 3 1) = (1, 2)

(2, 0) -------------------------- (4, -1) giving the graph below [red]

 

 

 

 

 

 

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