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

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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:

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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)

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