Translations textbook questions pg. 12-15 (1)
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Millwoods Christian School *
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Course
MISC
Subject
Mathematics
Date
Nov 24, 2024
Type
docx
Pages
10
Uploaded by CorporalThunder8499
Key Ideas
• Translations are transformations that shift all points on the graph of a function up, down
, left
, and right without changing the shape or orientation of the graph. • The table summarizes translations of the function y = f
(x
)
. Function y
-
k = f
(x) or Transformation
from y = f(x
) A vertical translation Mappin
g (x
, y
) → (x, y + k
)
Example y = f
(
x
) + k
If k > 0
, the translation is up. If k < 0
, the translation is down
. y
-
k
=
f
(x
)
, k
> 0, ▼y = f
(
x) y
-
k= f
(
x)
, k
<0 y = f(x - h) A horizontal translation (x, y)
(x
+
h
, y) If h > 0
, the translation is to the y
= f(x
-
h)
, h
>0 right
. y = f(x
) If h
< 0
, the translation is to the left
. y = f(x − h)
, h
<0
A sketch of the graph of y - k = f(
x − h
)
, or y = f
(x − h) + k, can be created by translating key points on the graph of the base function y = f
(x)
. Check Your Understanding Practise 1. For each function
, state the values of h and k
, the parameters that represent the horizontal and vertical translations applied to y = f(x
). -
a) y − 5 = f(x
) b
) y = f
(
x
) — 4 2. Given the graph of y = f
(
x
) and each of the following transformations, • state the coordinates of the image points A'
, B
'
, C'
, D' and E' • sketch the graph of the transformed function a) g(x) = f
(
x
) + 3 b
) h(x
) = f
(
x − 2) c) s(
x
) = f
(x + 4
) d) t(
x) = f
(x
) − 2 c) y = f(
x + 1) d) y + 3 = f(
x − 7) e) y = f(
x + 2
) + 4 12 MHR Chapter 1 YA y
= f(x
)
2 BC -2 0 DI TE 2 * X ய
A
3. Describe
, using mapping notation
, how the graphs of the following functions can be obtained from the graph of y = f(x
). a) y = f(
x + 10
) b
) y + 6 = f(
x
) c
) y = f(x − 7) + 4 - d) y 3 = f(
x − 1) - 4. Given the graph of y = f
(
x
), sketch the graph of the transformed function
. Describe the transformation that can be applied to the graph of f
(
x
) to obtain the graph of the transformed function
. Then
, write the transformation using mapping notation. a) r
(
x) = f
(x + 4
) – 3 8. Copy and complete the table
. Transformed
Function Transformation of
Points (x, y) → (x
, y+5) Translation vertical y = f
(x
) + 5 y = f
(
x + 7) (x, y
) → (x − 7, y) y = f
(x
-
3) y = f(x
) - 6 y+9 = f(x
+4
) horizontal and vertical horizontal and vertical b
) s
(x) = f
(
x − 2) — 4 c) t
(x
) = f
(x − 2) + 5 A Ул DE 21
N
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d) v(x) = f
(
x + 3) + 2 y = f(
x) B C -6 -4 -2 0 -2-
Apply
N -N AX 5. For each transformation, identify the values of h and k
. Then, write the equation of the transformed function in the form y - k = f
(
x − h
)
. 1 - a) f
(x) = 1, translated 5 units to the left and 4 units up b) f
(x) = x2
, translated 8 units to the right and 6 units up c) f
(
x
) = |
x|
, translated 10 units to the right and 8 units down d) y = f(x
), translated 7 units to the left and 12 units down 6. What vertical translation is applied to =
x2 if the transformed graph passes
through the point (
4
, 19
)? y 7. What horizontal translation is applied to y = x2 if the translation image graph passes through the point (
5
, 16)? horizontal and vertical y = f(x − h) + k (
x, y) → (
x + 4, y — 6)
(x, y)
(
x
-
2
, y+3) 9. The graph of the function y = x2 is translated 4 units to the left and 5 units up to form the transformed function y = g(x)
. a
) Determine the equation of the function y = g(
x
)
. b) What are the domain and range of the image function
? c) How could you use the description of the translation of the function y = x2 to
determine the domain and range of the
image function? 10. The graph of ƒ
(x
) = |x| is transformed to the graph of g
(
x) = f
(
x − 9
) + 5
. a) Determine the equation of the function g
(
x
)
. b
) Compare the graph of g(x
) to the graph of the base function f
(
x
)
. c
) Determine three points on the graph of f(
x
)
. Write the coordinates of the image points if you perform the horizontal translation
first and then the vertical translation
. d) Using the same original points from part c
)
, write the coordinates of the image points if you perform the vertical translation first and then the horizontal translation
. e) What do you notice about the coordinates of the image points from parts c
) and d)? Is the order of the translations important
? 1.1 Horizontal and Vertical Translations MHR 13 ⚫
11. The graph of the function drawn in red is a translation of the original function drawn in blue
. Write the equation of the translated function in the form y − k = f
(
x − h
)
. a)
У
41 - 13. Architects and designers often use translations in their designs. The image shown is from an Italian roadway
. 2
1
|
f(
x
) = X
· 4 6 Χ b)
У 4 M -2 0 -2-
-4- y = f
(
x)
-N 2
4
-
L
O
6
a) Use the coordinate plane overlay with the base semicircle shown to describe the approximate transformations of the semicircles
. b
) If the semicircle at the bottom left of the image is defined by the function y = f
(
x
), state the approximate equations of three other semicircles
. 14. This Pow Wow belt shows a frieze
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pattern where a particular image has been translated throughout the length of the belt
. 12. Janine is an avid cyclist
. After cycling to a lake and back home
, she graphs her distance versus time (
graph A)
. a) If she left her house at 12 noon, briefly describe a possible scenario for Janine's trip. b) Describe the differences it would make to Janine's cycling trip if the graph of the function were translated
, as shown in graph B. c) The equation for graph A could be written as y = f
(
x
)
. Write the equation for graph B.
A Distance From Home (
km) 30 201 10 XE a) With or without technology, create a design using a pattern that is a function
. Use a minimum of four horizontal translations of your function to create your own frieze pattern
. b) Describe the translation of your design in words and in an equation of the form y = f(
x − h
)
. Did You Know? In First Nations communities today
, Pow Wows have evolved into multi
-
tribal festivals
. Traditional dances are performed by men
, women
, and children
. The dancers wear traditional regalia specific to their dance style and nation of origin
. 0
2 4
6
8
10 x Time (h) 14 MHR
Chapter ⚫
1 15. Michele Lake and Coral Lake
, located near the Columbia Ice Fields
, are the only two
lakes in Alberta in which rare golden trout live
. Suppose the graph represents the number of
golden trout in Michelle Lake in the years since 1970
. N
u
m
b
er
of
Tr
o
ut
(h
u
n
d
re
d
s)
f
(
t
)▲
20- 16 12
8-
4 0
2 4 6 8 10 Time Since 1970 (years) Let the function f(t
) represent the number of fish in Michelle Lake since 1970
. Describe an event or a situation for the fish population that would result in the following transformations of the graph. Then
, use function notation to represent the transformation
. a) a vertical translation of 2 units up b) a horizontal
translation of 3 units to the right 16. Paul is an interior house painter. He determines that the function n = f
(
A
) gives the number of gallons
, n
, of paint needed to cover
an area
, A
, in square metres. Interpret n = f(
A
) +
10 and n = f
(A
) + 10 and n = f(A + 10
) in this context
.
Extend 17. The graph of the function y = x2 is translated to an image parabola with zeros 7 and 1
. a) Determine the equation of the image function
. b) Describe the translations on the graph of y = x2
.
c) Determine the y
-
intercept of the translated function
. 18. Use translations to describe how the 1 X graph of y = compares to the graph
of each function
. 1 a) y 4 = X c) y 3 = X - 1 5
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1 b
) y = x
+
2 d
) y = 1 x + 3 4
19. a) Predict the relationship between the graph of y = x3- x2 and the graph of y
+ 3
(x
-
2)3 — (
x − 2)
2
. - b) Graph each function to verify your prediction
. Create Connections C1 The graph of the function y = f
(x
) is transformed to the graph of y = f(
x − h
) + k
. a) Show that the order in which you apply translations does not matter. Explain why this is true
. b) How are the domain and range affected by the parameters h and k? C2 Complete the square and explain how to transform the graph of y = x2 to the graph of each function
. a) f(
x
) = x2 + 2x + 1 b) g(x) = x2 4x + 3 C3 The roots of the quadratic equation x2 X 120 are -3 and 4
. Determine the roots of the equation (x − 5
)
2 – (x − 5) — 12 = 0
. C4 The function f(
x
) = x + 4 could be a vertical translation of 4 units up or a horizontal translation of 4 units to the left
. Explain why
. 1.1 Horizontal and Vertical Translations MHR ⚫
15