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Section 1.6 – Parent Functions and Intro to Transformations Objective – To recognize the graph and equation of the 8 parent functions and be able to analyze their graphs. To recognize basic transformations
Parent Functions Parent Function: Constant ! = #
Domain: (−∞,∞) Range: c Constant: (−∞,∞) No Relative Max/Min Even/Odd: Even
Parent Function: Linear ! = )
Domain: (−∞,∞) Range: (−∞,∞) Increasing: (−∞,∞) No Relative Max/Min Even/Odd: Odd
Parent Function: Quadratic ! = )*
Domain: (−∞,∞) Range: [0,∞) Increasing: (0,∞) Decreasing: (−∞,0) Relative Min: (0, 0) Even/Odd: Even
Parent Function: Square Root ! = √)
Domain: [0,∞) Range: [0,∞) Increasing: (0,∞) Relative Min: (0, 0) Even/Odd: Neither
Parent Function: Cubic ! = ).
Domain: (−∞,∞) Range: (−∞,∞) Increasing: (−∞,∞) No Relative Max/Min Even/Odd: Odd
Parent Function: Cube Root ! = √)3
Domain: (−∞,∞) Range: (−∞,∞) Increasing: (−∞,∞) No Relative Max/Min Even/Odd: Odd
Memorize
Parent Function: Absolute Value ! = |)|
Domain: (−∞,∞) Range: [0,∞) Increasing: (0,∞) Decreasing: (−∞,0) Relative Min: (0, 0) Even/Odd: Even
Parent Function: Reciprocal ! = 1 )2
Domain: (−∞, 0) ∪ (0,∞)
Range:
(−∞, 0) ∪ (0,∞) Decreasing:
(−∞, 0) ∪ (0,∞) No Relative Max/Min Even/Odd: Odd
A transformation is a general term for specific ways to manipulate the shape of a point, a line, or shape. The original shape of the object is called the pre-image and the final shape and position of the object is the image under the transformation. In mathematics, we can use transformations of a known graph to graph a function with a similar equation. This is achieved through translations (horizontal and vertical shifts), reflections (flips), and dilations (stretching and compressing) of the known graph. Translation: A rigid transformation that shifts a graph horizontally or vertically.
Vertical Shift (OUTSIDE the Parent) Horizontal Shift (INSIDE the Parent) Upward c units: f (x) + c Left c units: f (x + c)
Downward c units: f (x) – c Right c units: f (x – c) Parent f (x) = x2
Shift left 2: f (x) = (x + 2)2
Shift right 2: f (x) = (x – 2)2
Parent f (x) = x2
Shift up 2: f (x) = x2 + 2
Shift down 2: f (x) = x2 – 2
Reflection: A rigid transformation that performs a flip of the parent function f(x) over the x- or y-axis.
Reflection across the x-axis (Vertical Flip ~ OUTSIDE the Parent)
Reflection across the y-axis (Horizontal Flip ~ INSIDE the Parent)
– f (x) f (–x) Parent f(x) = x2
Reflected over x-axis: f(x) = – x2
Parent f(x) = x3
Reflected over y-axis: f(x) = (–x)3
-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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x
y
-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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x
y
-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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x
y
-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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-4 -3-2 -1 1 2 3 4 5
-4-3-2-1
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J2 J2
13
Dilation (or Scale Change): A non-rigid transformation that performs a vertical or horizontal stretch or compression in the graph of the parent function f (x). ***A stretch gets bigger and a compress gets smaller
Vertical Stretch & Compress (OUTSIDE the Parent)
Horizontal Stretch & Compress (INSIDE the Parent)
Changes the function f (x) to c · f (x) and a point (x, y) to (x, cy)
Changes the function f (x) to f (c · x) and a point (x, y) to
.
If c > 1, the graph stretches vertically away from the x-axis by a factor of c If c < 1, the graph compresses vertically towards the x-axis by a factor of c
If c > 1, the graph compresses horizontally towards the y-axis by a factor of
4 52 . If c < 1, the graph stretches horizontally away from the y-axis by a factor of 4 52 .
******Remember… Horizontal is always OPPOSITE and always INSIDE!!! Example: Identify the parent function that can be used to graph each function. Don’t graph. Then state the transformations. a) g (x) = │–4x│ Parent Function: Transformation: b) 6()) = 3√) + 5 Parent Function: Transformation:
c) h (x) = 1 – 5x3
Parent Function: Transformation:
d) h (x) = ½ (x – 4)2 – 1 Parent Function: Transformation:
( )yxc , 1
f x _IIa fCx 2 5
Mx 5 3 1
absolutevalue y 1 1 cubic y x3
flip every axis flip over x axishoriz comp of Ty Vert stretch of 5
up I
r c rsauareroot y Tx quadratic y ylefts right 4Vert stretch of 3 vert Camp of t2
down
Example: Use the graphs of f (x) below to graph the given transformations g (x) = 2· f (x)
g (x) = f (x + 2) + 3
g (x) = f (– x)
g (x) = f (2x)
yleft wp3
Cd or
fX z 0
HITE1.4 on 44
II
norfair
norifor
fOO 2 2
infnegatejinanse9.9 a
41942Innit
2 O Xo2 2 O
O 2I 2