WARM UP DESCRIBE THE TRANSFORMATION FROM F(X) TO G(X)

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2 WARM UP DESCRIBE THE TRANSFORMATION FROM F(X) TO G(X)

3 WHAT YOU WILL LEARN HOW TO GRAPH THE PARENT FUNCTIONS OF VARIOUS FUNCTIONS. HOW TO IDENTIFY THE KEY FEATURES OF FUNCTIONS. HOW TO TRANSFORM FUNCTIONS FROM THE PARENT FUNCTION.

4 There are several types of functions (linear, quadratic, absolute value, sine, cosine, exponential, cubic, etc.). Each of these can be considered a family with unique characteristics that are shared among the members. The parent function is the most basic function in each family. It is used to create more complicated functions.

5 LINEAR FUNCTION KEY FEATURES DOMAIN: RANGE: INTERCEPTS: Y INT 0, 0 X INT 0, 0 INCREASING/DECREASING: INC WHERE POSITIVE/NEGATIVE: NEG, 0 POS 0, RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: ODD END BEHAVIOR:

6 QUADRATIC FUNCTION KEY FEATURES DOMAIN: RANGE: 0, INTERCEPTS: Y INT 0, 0 X INT 0, 0 INCREASING/DECREASING: DEC WHERE POSITIVE/NEGATIVE: POS, 0 INC 0,, 0 0, RELATIVE MAXIMUMS/MINIMUMS: MINIMUM AT 0, 0 SYMMETRIES: EVEN END BEHAVIOR:

7 ABSOLUTE VALUE FUNCTION KEY FEATURES DOMAIN: RANGE: 0, INTERCEPTS: Y INT 0, 0 X INT 0, 0 INCREASING/DECREASING: DEC WHERE POSITIVE/NEGATIVE: POS, 0 INC 0,, 0 0, RELATIVE MAXIMUMS/MINIMUMS: MINIMUM AT 0, 0 SYMMETRIES: EVEN END BEHAVIOR:

8 CUBIC FUNCTION KEY FEATURES DOMAIN: RANGE: INTERCEPTS: Y INT 0, 0 INCREASING/DECREASING: WHERE POSITIVE/NEGATIVE: X INT 0, 0 INC NEG, 0 POS 0, RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: ODD END BEHAVIOR:

9 SQUARE ROOT FUNCTION KEY FEATURES DOMAIN: RANGE: 0, 0, INTERCEPTS: Y INT 0,0 INCREASING/DECREASING: WHERE POSITIVE/NEGATIVE: X INT 0, 0 INC 0, POS 0, RELATIVE MAXIMUMS/MINIMUMS: MINIMUM AT 0, 0 SYMMETRIES: NONE END BEHAVIOR:

10 CUBE ROOT FUNCTION KEY FEATURES DOMAIN: RANGE: INTERCEPTS: Y INT 0,0 INCREASING/DECREASING: WHERE POSITIVE/NEGATIVE: X INT 0, 0 INC NEG, 0 RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: ODD END BEHAVIOR: POS 0,

11 EXPONENTIAL FUNCTION KEY FEATURES DOMAIN: RANGE: 0, INTERCEPTS: Y INT 0,1 INCREASING/DECREASING: WHERE POSITIVE/NEGATIVE: INC POS RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: NONE END BEHAVIOR:

12 LOGARITHMIC FUNCTION KEY FEATURES DOMAIN: 0, RANGE: INTERCEPTS: X INT 1, 0 INCREASING/DECREASING: WHERE POSITIVE/NEGATIVE: INC 0, NEG 0,1 RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: NONE END BEHAVIOR: POS 1,

13 SINE FUNCTION KEY FEATURES DOMAIN: RANGE: 1, 1 INTERCEPTS: X INT INCREASING/DECREASING:,0 Y INT 0, 0 ALTERNATING INC AND DEC IN PERIODIC WAVES WHERE POSITIVE/NEGATIVE: ALTERNATING POS AND NEG IN PERIODIC WAVES RELATIVE MAXIMUMS/MINIMUMS: ABSOLUTE MAXIMUM OF 1 ABSOLUTE MINIMUM OF -1 SYMMETRIES: ODD END BEHAVIOR: NONE VALUES OSCILLATE BETWEEN -1 AND 1 AND APPROACH NO LIMIT

14 COSINE FUNCTION KEY FEATURES DOMAIN: RANGE: 1, 1 INTERCEPTS: X INT INCREASING/DECREASING:, 0 WHERE K IS ODD Y INT 0, 1 ALTERNATING INC AND DEC IN PERIODIC WAVES WHERE POSITIVE/NEGATIVE: ALTERNATING POS AND NEG IN PERIODIC WAVES RELATIVE MAXIMUMS/MINIMUMS: ABSOLUTE MAXIMUM OF 1 ABSOLUTE MINIMUM OF -1 SYMMETRIES: EVEN END BEHAVIOR: NONE VALUES OSCILLATE BETWEEN -1 AND 1 AND APPROACH NO LIMIT

15 STEP FUNCTION KEY FEATURES DOMAIN: RANGE: INTERCEPTS: X INT!!"# #$% INCREASING/DECREASING: 0,1!& NEITHER INC OR DEC WHERE POSITIVE/NEGATIVE: NEG. 0 POS 1, RELATIVE MAXIMUMS/MINIMUMS: NONE SYMMETRIES: NONE END BEHAVIOR: 0 Y INT 0, 0

16 TRANSFORMATIONS VERTICAL SHIFT HORIZONTAL SHIFT REFLECTIONS Y AXIS VERTICAL STRETCH VERTICAL STRETCH (SHRINK) HORIZONTAL STRETCH (SHRINK) ( X-AXIS 2 SHIFT DOWN 2 3 SHIFT RIGHT 3 VERTICAL STRETCH BY A FACTOR 2 OF 2 VERTICAL STRETCH BY A FACTOR )1 OF 0* *1, - 3 FACTOR OF,. HORIZONTAL STRETCH BY A

17 HORIZONTAL STRETCH THE TRANSFORMED FUNCTION HAS TWICE AS MANY CYCLES AS THE ORIGINAL (2X). THE X VALUES OF THE GRAPH ARE ½ OF THE PARENT FUNCTION. THUS, THIS IS A HORIZONTAL STRETCH BY A FACTOR OF 1/2.

18 EXAMPLE - GRAPH F(X), GRAPH EACH NEW FUNCTION, WITHOUT TECHNOLOGY, AND DESCRIBE THE TRANSFORMATION. DETERMINE IF THE TRANSFORMED FUNCTION IS EVEN, ODD OR NEITHER. G(X) = F(-X) H(X)=2F(X+1)

19 EXAMPLE - GRAPH F(X), GRAPH EACH NEW FUNCTION, WITHOUT TECHNOLOGY, AND DESCRIBE THE TRANSFORMATION. DETERMINE IF THE TRANSFORMED FUNCTION IS EVEN, ODD OR NEITHER. G(X) = 1/2F(X) H(X)=F(2X)-1

20 EXAMPLE DESCRIBE THE TRANSFORMATION AND DETERMINE IF THE TRANSFORMATION IS EVEN OR ODD.

21 EXAMPLE DESCRIBE THE TRANSFORMATION AND DETERMINE IF THE TRANSFORMATION IS EVEN OR ODD.

22 EXAMPLE DETERMINE IF THE FUNCTION CAN BE OBTAINED BY THE PARENT FUNCTION BY LOOKING AT THE GRAPH YOU CAN SEE IT HAS THREE REAL ROOTS WHAT ABOUT THE PARENT FUNCTION? THIS FUNCTION IS INCREASING PART OF THE TIME AND DECREASING PART OF THE TIME WHAT ABOUT THE PARENT FUNCTION? IT IS NOT POSSIBLE TO OBTAIN G(X) THROUGH BASIC TRANSFORMATIONS OF THE PARENT FUNCTION.

23 EXAMPLE DETERMINE IF THE FUNCTION CAN BE OBTAINED BY THE PARENT FUNCTION BY LOOKING AT THE GRAPH YOU CAN SEE IT HAS BEEN REFLECTED OVER THE X-AXIS, MOVED UP ONE, AND MOVED TO THE RIGHT THREE. IF THAT WAS THE CASE THE EQUATION WOULD BE 3 1 MULTIPLY AND SIMPLIFY THE EQUATION ABOVE. WHAT DO YOU GET? THIS FUNCTION CAN BE OBTAINED BY THE PARENT FUNCTION.

24 EXAMPLE DETERMINE THE TRANSFORMATIONS THAT WERE USED TO CHANGE THE PARENT FUNCTION TO THE FUNCTION GRAPHED. REFLECTION OVER X-AXIS VERTICAL STRETCH BY A FACTOR OF 1/3 TRANSLATED UP 5 UNITS TRANSLATED RIGHT 4 UNITS

25 EXAMPLE DETERMINE THE TRANSFORMATIONS THAT WERE USED TO CHANGE THE PARENT FUNCTION TO THE FUNCTION GRAPHED. TRANSLATED UP 4 UNITS TRANSLATED LEFT 1 UNIT