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Hon Algebra 2: Unit 1
GRAPHING FUNCTIONS and TRANSFORMATIONS
For any function there are 4 basic ways to transform the shape of its graph. The original function f(x) is often called the parent function and has specific properties and key points to assist in graphing.
1) Vertical Translation (Shift): Graph is moved ______________________________________
2) Horizontal Translation (Shift): Graph is moved __________________________________________
3) Vertical Dilations, Contractions, and Reflections: In the vertical direction, __________________________________________________________________
4) Horizontal Dilations, Contractions, and Reflections: In the horizontal direction, _________________________________________________________________
Identify the points of the given parent function f(x) in the graph:
· Graph each transformation of the parent function and describe the change from the original.
x
y or f(x)
Identify the points of the given parent function g(x) in the graph:
· Graph each transformation of the parent function and describe the change from the original.
How do operations in different locations of the parent function create transformations?
1) Vertical Translation (Shift):
a. UP:
b. DOWN:
2) Horizontal Translation (Shift):
a. LEFT:
b. RIGHT:
3) Vertical Dilations, Contractions, and Reflections:
a. STRETCH:
b. SHRINK:
c. FLIP:
GENERAL FORM FOR TRANSFORMATIONS of FUNCTION f(x): a • f(x – h) + k
“h” = horizontal shift
“k” = vertical shift
“a” = vertical dilation, contraction, and reflection
DESCRIBE THE TRANSFORMATIONS FOR THE GIVEN EXPRESSIONS
For parent functions f(x), g(x), or h(x)
1) f(x – 1) + 2
2) h(x + 7) + 8
3) 2f(x – 1)
4) -3 f(x) + 2
5) ½ g(x) – 9
6) -3/4h(x + 6)
7) 2f(x + 3) – 5
8) –g(x – 4) + 7
9) 2/3h(x + 1) + 5
SPECIFIC FUNCTIONS AND THEIR TRANSFORMATIONS
ABSOLUTE VALUE:
· Parent Function: f(x) = |x|
· Transformation Function:
· Important Point: (h, k)
· Generic Shape:
· DOMAIN:
· RANGE:
QUADRATIC:
· Parent Function: f(x) = x2
· Transformation Function:
· Important Point: (h, k)
· Generic Shape:
· DOMAIN:
· RANGE:
PRACTICE SHIFTS WITH ABSOLUTE AND QUADRATIC FUNCTIONS
Section 1: Graph
1)
|
4
|
+
=
x
y
2)
5
|
|
-
=
x
y
3)
2
|
3
|
+
-
=
x
y
4)
|
|
3
x
y
=
5)
4
|
|
2
1
+
-
=
x
y
6)
2
x
y
-
=
7)
6
2
-
=
x
y
8)
2
)
3
(
-
=
x
y
9)
1
)
2
(
2
+
+
=
x
y
Section 2: Based on each function statement describe the transformations from the parent.
1)
5
|
|
+
=
x
y
2)
|
2
|
+
=
x
y
3)
|
9
|
-
=
x
y
4)
|
|
4
x
y
-
=
5)
3
|
|
3
-
=
x
y
6)
|
5
|
3
1
-
=
x
y
7)
3
|
2
|
-
+
=
x
y
8)
7
|
6
|
2
-
+
=
x
y
9)
1
2
|
8
|
+
-
-
=
x
y
10)
3
2
+
=
x
y
11)
4
2
-
-
=
x
y
12)
2
)
5
(
+
=
x
y
13)
2
)
7
(
2
-
=
x
y
14)
7
)
2
(
2
-
-
=
x
y
15)
2
)
4
(
3
1
+
-
=
x
y
16)
(
)
1
9
3
2
+
-
=
x
y
17)
3
)
6
(
3
2
2
-
+
=
x
y
18)
2
5
)
4
(
7
2
-
-
=
x
y
Section 2 Exploration: Determine for the pair functions what transformations are occurring from the first to second function overall.
1)
5
|
|
+
=
x
y
and
9
|
|
+
=
x
y
2)
|
3
|
-
=
x
y
and
|
1
|
+
=
x
y
3)
4
|
2
|
+
-
=
x
y
and
4
|
|
+
=
x
y
4)
7
|
8
|
3
+
+
=
x
y
and
4
|
8
|
3
-
+
=
x
y
5)
6
|
|
3
1
+
=
x
y
and
3
|
6
|
3
1
+
-
=
x
y
6)
1
|
3
|
-
+
=
x
y
and
9
|
5
|
-
-
=
x
y
7)
3
2
+
=
x
y
and
2
2
-
=
x
y
8)
2
)
4
(
-
=
x
y
and
2
)
7
(
-
=
x
y
9)
5
)
1
(
2
2
+
+
=
x
y
and
2
)
1
(
2
+
=
x
y
10)
2
2
+
=
x
y
and
2
)
5
(
2
+
+
=
x
y
11)
2
)
4
(
-
=
x
y
and
9
)
4
(
2
+
+
=
x
y
12)
2
)
3
(
2
+
-
-
=
x
y
and
6
)
8
(
2
-
-
-
=
x
y
Section 3: Write the EQUATIONS with described shifts and given parent functions.
1) y = |x|; Up 7 and Left 3
1. _____________________________
2) y = x2; Reflects and Right 9
2. _____________________________
3) y = |x|; Down 4 and Right 1
3. _____________________________
4) y = x2; Down 2, Reflects, Vertical shrink of 1/6 4. _____________________________
5) y = |x|; Right 6, Vertical stretch of 2
5. _____________________________
6) y = x2; Left 5/3, Up 7/12, Vertical stretch of 4/36. _____________________________
7) y = |x|; Right 9 and Down 2
7. _____________________________
8) y = x2; Vertical Shrink of ½ and Up 3
8. _____________________________
9) y = |x|; Left 6 and Reflects
9. _____________________________
10) y = x2; Down 6, Vertical Stretch of 5, Right 410. _____________________________
11) y = |x|; Reflects, Up 2 and Left 9
11. _____________________________
12) y = x2; Vertical Shrink 3/7, Right 1/2, Down 7/912. _____________________________
SQUARE ROOT:
· Parent Function:
x
x
f
=
)
(
· Transformation Function:
· Important Point: (h, k)
· Generic Shape:
· DOMAIN:
· RANGE:
CUBIC: “ODD FUNCTION”
· Parent Function: f(x) = x3
Transformation Function:
· Important Point: (h, k)
· Generic Shape:
· DOMAIN:
· RANGE:
PRACTICE SHIFTS WITH CUBE AND SQUARE ROOT FUNCTIONS
Section 1: Graph
1)
4
+
=
x
y
2)
3
-
=
x
y
3)
3
2
+
-
=
x
y
4)
x
y
3
=
5)
5
+
-
=
x
y
6)
3
2
2
-
+
-
=
x
y
7)
3
x
y
-
=
8)
3
3
-
=
x
y
9)
3
)
4
(
-
=
x
y
10)
3
)
1
(
2
+
=
x
y
Section 2: Based on each function statement describe the transformations from the parent.
1)
6
-
=
x
y
2)
(
)
3
4
+
=
x
y
3)
x
y
2
=
4)
(
)
3
5
4
x
y
-
=
5)
6
3
-
-
=
x
y
6)
5
7
+
+
=
x
y
7)
1
4
7
-
-
=
x
y
8)
(
)
3
8
3
-
-
=
x
y
9)
7
)
5
(
2
3
+
+
=
x
y
10)
2
4
3
-
-
-
=
x
y
Section 2 Exploration: Determine for the pair functions what transformations are occurring from the first to second function overall.
1)
7
+
=
x
y
and
4
+
=
x
y
2)
(
)
1
1
3
-
-
=
x
y
and
(
)
2
8
3
+
-
=
x
y
3)
5
3
+
=
x
y
and
1
3
+
=
x
y
4)
3
2
+
-
=
x
y
and
4
3
-
+
=
x
y
Section 3: Write the EQUATIONS with described shifts and given parent functions.
1)
x
y
=
; Down 4 and Right 2
1. _____________________________
2) y = x3; Reflects and Right 3
2. _____________________________
3)
x
y
=
; Vertical Shrink 2/5, Left 7
3. _____________________________
4) y = x3; Down 2, Reflects, Vertical Stretch 4 4. _____________________________
5)
x
y
=
; Reflect, Vertical stretch of 3, Up 65. _____________________________
6) y = x3; Vertical Shrink 2/3, Left 9
6. _____________________________
7)
x
y
=
; Vertical Stretch 5, Down 7, Right 37. _____________________________
8) y = x3; Vertical Shrink of ½, Left 2, Up 8
8. _____________________________
GENERAL PRACTICE:
PART 1: For each of the given graphs, write the EQUATION that would create that graph.
· Graphs are approximately drawn to scale
· There are NO Vertical Shrinks or Stretches from the parent function.
· Focus on the important point of each function based on its parent function.
Section 2: (1) Graph the transformation (2) Label 3 points guaranteed to be on the graph
1)
2
|
3
|
-
+
=
x
y
2)
4
2
+
-
=
x
y
3)
1
3
-
=
x
y
4)
(
)
3
3
3
-
-
=
x
y
5)
4
|
5
|
3
2
+
-
=
x
y
6)
3
)
4
(
2
2
+
+
=
x
y
Section 2a: Identify the DOMAIN and RANGE for each graph:
Section 3: Identify the transformations of each listed function and name the parent function
1)
5
4
-
+
=
x
y
2)
(
)
3
2
3
+
=
x
y
3)
8
3
2
2
+
=
x
y
4)
(
)
7
2
4
5
3
+
-
-
=
x
y
5)
3
4
1
-
=
x
y
6)
2
3
+
+
-
=
x
y
7)
(
)
4
6
2
2
-
-
=
x
y
8)
2
3
3
8
-
+
=
x
y
9)
3
)
1
(
2
-
-
=
x
y
10)
2
4
7
3
-
-
-
=
x
y
Section 4: Write the equation from the given parent function and transformations
· List the coordinate for the new “important point” after transformation
1) Quadratic; Up 3 and Left 7
2) Absolute; Reflects and Right 2
3) Cube; Down 4 and Right 1
4) Square Root; Down 2, Reflects, Shrink 1/6
5) Cube; Right 6, Stretch 2
6) Cube; Left 1, Up 1, Stretch of 4/3
7) Absolute; Right 9 and Down 2
8) Square Root; Vertical 2, Right 3, Up 2
9) Square Root; Vertical 2, Left 2, Down 3
10) Cube; Down 6, Stretch of 5, Right 4
11) Absolute; Reflects, Up 2 and Left 9
12) Quadratic; Shrink 3/4, Down 2
13) Quadratic; Reflects, Stretch 4/3, Left 2
14) Square Root; Right 1 and Up 3
f(x) – 2
f(x) + 3
f(x)
f(x + 1)
-1•f(x)
2•f(x)
f(x – 2)
½•f(x)
g(x) + 1
g(x) – 4
g(x)
g(x + 3)
-2•g(x)
3•g(x)
g(x + 1)
g(x – 3)
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