Transformational Geometry
y  ax 2  bx  c
y  a ( x  h) 2  k
1
( y  k )  ( x  h) 2
a
x, y   ( x  h, ay  k )
General Form

Standard Form

Transformational Form

Mapping Notation
Mathematics 3204
1.
2.
Public Exam Questions
Transformational Geometry
Which quadratic function has the widest graph?
(A)
(B)
2( y  3)  ( x  4)2
2
1
2 ( y  3)  ( x  4)
(C)
(D)
2( y  3)  ( x  4) 2
5( y  3)  ( x  4) 2
The graph of y  x 2 has been transformed as shown so that its new vertex is
(3,1) . What is the vertical stretch factor?
y
(2,1)
(1,-1)
(A)
(B)
(C)
(D)
3.
4.
(3,-1)
x
1
2
1
2
3
The graph of y  x 2 has been transformed using the mapping rule
( x, y )  ( x  1, 2 y  3) . What is the new function?
(A)
(B)
(C)
2 y  3  ( x  1) 2
2( y  3)  ( x  1)2
 12 ( y  3)  ( x  1) 2
(D)
1
2
( y  3)  ( x  1) 2
What is the general form of the function y  ( x  3)2  11 ?
(A)
(B)
(C)
(D)
y  11  ( x  3)2
y  x 2  20
y  x2  6 x  2
y  x 2  6 x  20
14
Mathematics 3204
5.
7.
8.
Transformational Geometry
What equation is produced if y  x 2 is translated 4 units down, 3 units right, and
vertically stretched by a factor of 10?
(A)
1
10
( y  4)  ( x  3) 2
(B)
1
10
( y  4)  ( x  3) 2
(C)
(D)
6.
Public Exam Questions
10( y  4)  ( x  3) 2
10( y  4)  ( x  3)2
Which has the smallest vertical stretch factor compared to y  x 2 ?
(A)
 52 ( y  1)  ( x  1) 2
(B)
 23 ( y  1)  ( x  1) 2
(C)
(D)
2( y  1)  ( x  1)2
3( y  1)  ( x  1)2
What is the standard form of
(A)
(B)
y  2( x  1)2  3
y   12 ( x  1) 2  3
(C)
y  12 ( x  1) 2  3
(D)
y  2( x  1)2  3
1
2
( y  3)  ( x  1) 2 ?
10
2
4
6
8
– 2
1– 2
2
3
4
5
6
7
1
Which is the quadratic function for the graph shown?
y
10
8
(A)
(B)
(C)
(D)
( y  1)  ( x  3)
( y  1)  ( x  3) 2
( y  1)  ( x  3)2
( y  1)  ( x  3)2
2
6
4
2
– 2 – 1
– 2
15
1
2
3
4
5
6
7
x
Mathematics 3204
9.
Transformational Geometry
Which transformation of y  x 2 produces 2( y  3)  ( x  5)2 ?
(A)
vertically stretched by a factor of 2, translated horizontally 5 units left, and
translated vertically 3 units up
vertically stretched by a factor of 2, translated horizontally 5 units right,
and translated vertically 3 units down
vertically stretched by a factor of 12 , translated horizontally 5 units left,
and translated vertically 3 units up
vertically stretched by a factor of 12 , translated horizontally 5 units right,
and translated vertically 3 units down
(B)
(C)
(D)
10.
Public Exam Questions
The graph of y  x 2 is transformed using
1– 5
2
3
4
5
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
results?
y
(A)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
1
2
3
4
5
x
1
2
3
4
5
x
1
2
3
4
5
x
1
2
3
4
5
x
– 2
– 3
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
– 4
– 5
y
(B)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
– 4
– 5
y
(C)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
(D)
– 4
– 5
y
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
– 5
16
 x, y    x  2,  y 1 . Which graph
Mathematics 3204
11.
13.
Transformational Geometry
If y  x 2 is transformed to y  4( x  3)2  2 , what is the vertical stretch factor?
(A)
(B)
(C)
(D)
12.
Public Exam Questions
4
 14
1
4
4
What is the transformational form of y  x 2  12 x  3 ?
(A)
(B)
y  3  ( x  6)2
(C)
y  39  ( x  6)2
(D)
y  39  ( x  6)2
y  3  ( x  6)2
2
1– 5
2
3
4
5
1of y  x to the graph
2
3
4
Which mapping rule was used to transform the graph
1– 4
2
3
4
5
6
7
1
2
3
shown?
y
(A)
5
 x, y    x  3,  y  2
4
3
(B)
(C)
 x, y     x  3, y  2
1
– 4– 3– 2– 1
– 1
 x, y     x  3, y  2
– 2
– 3
(D)
14.
 x, y    x  3,  y  2
– 4
– 5
What is the mapping rule that will transform y  x 2 into the equation
3( y  5)  ( x  1)2 ?
(A)
(B)
(C)
(D)
(3,2)
2
 x, y    x 1, 13 y  5
 x, y    x 1, 13 y  5
 x, y    x  1,3 y  5
 x, y    x  1,3 y  5
17
1
2
3
4
5
6
7 x
Mathematics 3204
15.
Public Exam Questions
Transformational Geometry
The
graph of y  x 2 is transformed to ( y  1)  ( x  3)2 . Which graph results?
1– 5
2
3
4
5
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
y
(A)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
1
2
3
4
5 x
1
2
3
4
5 x
1
2
3
4
5 x
1
2
3
4
5 x
– 2
– 3
– 4
– 5
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
y
(B)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
– 5
y
(C)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
– 5
y
(D)
5
4
3
2
1
– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
– 5
16.
What is the vertical stretch factor in 0.5( y  2)  x 2 compared to y  x 2 ?
(A)
(B)
(C)
(D)
2
0.5
0.5
2
18
Mathematics 3204
17.
Public Exam Questions
Transformational Geometry
What is the transformational form of y   12 ( x  3) 2  7 ?
(A)
(B)
2( y  7)  ( x  3)2
 12 ( y  7)  ( x  3) 2
(C)
1
2
(D)
2( y  7)  ( x  3)2
( y  7)  ( x  3) 2
1– 2
2
3
4
5
6
7
1
18.
1
2
3
4
5
Which translation of y  x21– would
generate the graph shown?
y
7
6
5
4
3
2
1
– 5 – 4 – 3 – 2 – 1
– 1
– 2
(A)
(B)
(C)
(D)
19.
1
2
x
vertical translation of 4 , horizontal translation of 2
vertical translation of 4, horizontal translation of 2
vertical translation of 4, horizontal translation of 2
vertical translation of 4 , horizontal translation of 2
Which describes the graph of ( y  5)  ( x  1)2 compared to the graph of
y  x2 ?
(A)
(B)
(C)
(D)
reflection in the x-axis followed by vertical translation of
horizontal translation of 1
reflection in the x-axis followed by vertical translation of
translation of 1
reflection in the y-axis followed by vertical translation of
horizontal translation of 1
reflection in the y-axis followed by vertical translation of
translation of 1
19
5 and
5 and horizontal
5 and
5 and horizontal
Mathematics 3204
20.
Public Exam Questions
The graph of y  x 2 has been transformed using the mapping
 x, y  642–642x  4, 12 y  3 . Which graph represents the new quadratic?
2– 8
4
6
8
2
4
6
y
(A)
6
4
2
– 8– 6– 4– 2
– 2
2
– 4
6– 6
2
4
2
4
2– 8
4
6
8
2
4
6
4
6
8 x
(6, -1)
(4, -3)
– 6
y
(B)
6
4
2
– 8– 6– 4– 2
– 2
2
– 4
4
6 8 x
(5, -1)
(4, -3)
– 6
6– 6
2
4
2
4
2– 8
4
6
8
2
4
6
y
(C)
6
(-6, 5)
4
(-4, 3)
2
– 8– 6– 4– 2
– 2
2
4
6
8 x
– 4
– 6
6– 6
2
4
2
4
2– 8
4
6
8
2
4
6
(D)
y
6
(-5, 5)
4
(-4, 3)
2
– 8– 6– 4– 2
– 2
2
4
6
8 x
– 4
– 6
21.
Transformational Geometry
What is the standard form of 4( y  6)  ( x  1)2 ?
(A)
y  4 x 2  8 x  10
(B)
y   14 x2  12 x  25
4
(C)
(D)
y  4( x  1)2  6
y   14 ( x  1)2  6
20
Mathematics 3204
22.
Public Exam Questions
Transformational Geometry
2– 8
4
6
8
2
4
6
2– 8
4
6
8
2
4
6
The graph shown is transformed through a vertical translation
of 3 followed by a
reflection in the x-axis. Which represents the transformed graph?
6– 6
2
4
2
4
2– 8
4
6
8
2
4
6
y
8
y
6
(A)
6
4
2
4
– 8– 6– 4– 2
– 2
8– 8
2
4
6
2
4
6
2– 8
4
6
8
2
4
6
2
4
6
8 x
2
– 4
– 6
– 8– 6– 4– 2
– 2
y
8
(B)
6
– 4
4
2
– 8– 6– 4– 2
– 2
2
4
6
– 6
8 x
– 8
– 4
8– 8
2
4
6
2
4
6
2– 8
4
6
8
2
4
6
2
– 6
– 8
y
8
(C)
6
4
2
– 8– 6– 4– 2
– 2
2
4
6
8 x
2
4
6
8 x
– 4
8– 8
2
4
6
2
4
6
2– 8
4
6
8
2
4
6
– 6
– 8
y
(D)
8
6
4
2
– 8– 6– 4– 2
– 2
– 4
– 6
– 8
23.
Compared to the graph of y  x 2 , which quadratic has a graph with a vertical
stretch factor of 13 ?
(A)
(B)
( y  3)  x 2
( y  13 )  x 2
(C)
3 y  x2
2
1
3 y  x
(D)
24.
Which quadratic equation represents the transformation of y  x 2 under the
mapping rule  x, y    x  3,  12 y  1 ?
(A)
 12 ( y  1)  ( x  3) 2
(B)
(C)
2( y  1)  ( x  3)2
2
1
2 ( y  1)  ( x  3)
(D)
2( y  1)  ( x  3) 2
21
4
6
8 x
Mathematics 3204
25.
26.
27.
Public Exam Questions
What is the standard form of 4( y  1)  ( x  3)2 ?
(A)
y  14 ( x  3)2  1
(B)
y  14 ( x  3)2  1
(C)
(D)
y  4( x  3)2  1
y  4( x  3)2  1
What is the transformational form of y  x 2  8 x ?
(A)
( y  64)  ( x  4)2
(B)
(C)
(D)
( y  16)  ( x  4)2
( y  16)  ( x  4)2
( y  64)  ( x  4)2
5– 5
1
2
3
4
1
2
3
4
1– 6
2
3
4
1
2
3
4
5
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
8– 2
1
2
3
4
5
6
7
1
1– 5
2
3
4
5
1
2
3
4
5– 5
1
2
3
4
1
2
3
4
1– 5
2
3
4
5
1
2
3
4
Which has the greatest vertical stretch factor compared to y  x 2 ?
y
y
y
5
8
5
4
4
7
4
3
3
6
3
2
2
5
2
1
1
4
1
2
3
4
5 x
– 6– 5– 4– 3– 2– 1
– 1
1
2
3
1
3
4 x
2
– 5– 4– 3– 2– 1
– 1
1
– 2
– 2
– 2
– 3
– 3
– 4
– 4
– 5– 4– 3– 2– 1
– 1
– 5
– 5
– 2
A
(A)
(B)
(C)
(D)
y
5
– 5– 4– 3– 2– 1
– 1
28.
Transformational Geometry
B
1
2
3
4
5 x
C
A
B
C
D
Which describes the graph of
1 ( y  4)  ( x  3) 2
2
compared to y  x 2 ?
(A)
vertex  3, 4 and vertical stretch factor
(B)
vertex  3, 4 and vertical stretch factor 2
(C)
vertex  3,  4  and vertical stretch factor
(D)
vertex  3,  4  and vertical stretch factor 2
22
1
2
1
2
– 3
– 4
– 5
D
1
2
3
4
5 x
Mathematics 3204
29.
Public Exam Questions
Transformational Geometry
Which432121–– 432154321graph represents 2( y  1)  ( x  3) 2 ?
y
4
(A)
3
2
1
– 5 – 4 – 3 – 2 – 1
– 1
1
2
x
4
5
x
1
2
x
5
6
x
– 2
– 3
– 4
4– 4
1
2
3
1
2
3
1– 2
2
3
4
5
1
y
4
(B)
3
2
1
– 2 – 1
– 1
1
2
3
– 2
– 3
4– 4
1
2
3
1
2
3
1– 6
2
1
2
3
4
5
– 4
y
4
(C)
3
2
1
– 6– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
4– 4
1
2
3
1
2
3
1– 2
2
3
4
5
6
1
y
4
(D)
3
2
1
– 2– 1
– 1
1
2
3
4
– 2
– 3
– 4
30.
5– 5
1
2
3
4
1
2
3
4
1– 7
2
3
1
2
3
4
5
6
Which mapping rule applied to y  x 2 would generate the graph shown?
y
5
(A)
(B)
(C)
(D)
31.
 x, y    x  3,  y  2
 x, y    x  3, y  2
 x, y    x  3,  y  2
 x, y    x  3, y  2
4
3
2
1
– 7– 6– 5– 4– 3– 2– 1
– 1
– 2
– 3
– 4
– 5
What is the standard form of y  x 2  12 x  8 ?
(A)
y  ( x  6)2  28
(B)
(C)
y  ( x  6)2  4
y  ( x  6)2  4
(D)
y  ( x  6)2  28
23
1
2
3 x
Mathematics 3204
Public Exam Questions
Transformational Geometry
Answers Transformational Geometry
1. D
23. C
2. C
24. B
3. C
25. B
4. D
26. C
5. B
27. A
6. D
28. D
7. D
29. B
8. D
30. A
9. D
31. A
10. D
11. D
12. C
13. D
14. B
15. A
16. D
17. A
18. B
19. B
20. C
21. B
22. A
24