Math-Essentials Unit 3 Review. Equations and Transformations of the Linear, Quadratic, Absolute Value, Square Root, and Cube Functions

Transcription

1 Math-Essentials Unit Review Equations and Transormations o the Linear, Quadratic, Absolute Value, Square Root, and Cube Functions

2 Vocabulary Relation: A mapping or pairing o input values to output values. Function: A relation where each input has exactly one output.

3 Is it a unction? input - -5 output 5 yes input - -5 output 5 yes

4 Is it a unction? input - -5 output 5 Relation only The input value has two outputs ( and 5 )

5 Is it a unction? input - -5 output 5 Not a unction nor a relation (There aren t any pairings.) input - -5 output 5 yes Each input has exactly one output (even though it s the same output or each).

6 6 ways to show a relation between inputs and outputs. Ordered Pairs: (, ), (, ), (-, ), etc. Data table: x - y Equation: y = x + 1 Function notation: () = Graph: Mapping input output -

7 For the linear unction: 1 1 Your Turn: Find: (0) (x) = -x + Find: (1) (0) = -(0) + = 0 + (0) = (0, ) (1) = -(1) + (1) = 1 = - + (1, 1)

8 What is equation o this line? Your Turn: 1 1

9 Your turn: Compare the equation and the table o values. x x x x - 0 y 8 y 6 y 6 y 6 The coeicient o x = x

10 Your turn: What is the equation that passes thru the data? mx b 5 x 1 ( 0) 1 mx 1 x x x - 0 y 6 1 y 5 y 5 The coeicient o x = y x 5

11 What is a parent unction Parent Function: The simplest unction in a amily o unctions (lines, parabolas, cubic unctions, etc.) ( )? ( 1)? ( 0)? ( 1)? ( )? x x y Build a table o values or the equation. Graph the ordered pairs. Up 1 Vertex: (0, 0) Right 1

12 Describe the transormations to the parent unction. ( x 1) ( 1)? (1) (1 1) (1) (0) ( 1) I you are not sure which way the let/right shit is, plug in the x-value o the vertex that you think it should be (in this case either +1 or -1). Vertex:(0, 0) Vertex:(1, )

13 I you are not sure which way the let/right shit is, plug in the x-value o the vertex that you think it should be. ( x 1) ( 1) ( 1)? ( 1) ( 11) ( 1) ( ) ( 1) () ( 1) 7 Vertex:(0, 0) Vertex:(-1, 7)?? Or Vertex: (1, )

14 y x y x Vertex: (0, ) Adding to the parent unction moves the graph up units. x x

15 Your Turn: Find the output value or the given input value: x (-5) =? ( 5) ( 5) ( 5) (5) ( 5) 9 ( x ) 6 () =? () ( ) () ( 1) ( ) (1) ( ) 6 6 6

16 Let s generalize the transormations x y ( 1) a( x h ) k Relection across x-axis vertical stretch actor Translates let/right translating up or down y ( x ) Relected across x-axis, vertically stretched by a actor o, translated up, translated right

17 Vertically stretched by a actor o, translated right 1 (or let 1?) x g ( x 1) g( 1)? g( 1) (1 1) g( 1) (0) g( 1) 0 g( 1)? g( 1) ( 11) g( 1) ( ) right 1 g( 1) g( 1) () 8

18 Absolute Value Function x Build a table o values or each equation or domain elements: -, -1, 0, 1,. x y - Up Right 1 1 1

19 What are the transormations to the parent unction? x Relected across x-axis VSF = up Let (or right?) (-) =? ( ) ( ) ( ) 0 ( ) y x

20 What are the transormations to the parent unction? x Relected across x-axis VSF = up Let (or right?) () =? ( ) () ( ) ( ) 1 ( ) 8 y x Let

21 y x y ( 1) a( x h ) k Relection across x-axis Vertical stretch actor Shit let/right shit up or down y x y ( 1) a x h k y ( x 5) VSF=, translated let 5, translated down 5 x relected across x-axis VSF=5, translated right translated up

22 Your Turn: Find the output value or the given input value: x () =? ( ) ( ) 0 ( ) x (-) =? ( ) ( ) 0 ( )

23 Square Root Function x Up 1 Right 1 Right End Point: (0, 0) Up Build a table o values or each equation or domain elements: -, -1, 0, 1,. x 9 y ?? This is the irst unction, so ar, that does NOT have all real numbers as the domain.

24 Describe the transormation to the parent unction: y x Up, right y x y x Down, relected across x-axis, VSF= let y ( 1) a x h k

25 Up 1 Right 1 End Point: (, ) y ( x )

26 Right 1 End Point: (-1, ) Down y ( x 1)

27 x ( )? ( ) ( ) ( ) 0 g x g( )? g( ) g( ) 6 y ( 1) a x h k

28 What shape does it have? x Build a table o values or each equation or domain elements: -, -1, 0, 1,. x y (-)= () What does this mean? ( )( )( ) 8

29 How can you tell i it has been vertically stretched? x Not vertically stretched: right 1, up 1 From the anchor point ( 1) a( x h ) k

30 x What is the equation o the graph below? x

31 x What is the equation o the graph below? x

32 x What is the equation o the graph below? ( x 1) 1

33 x What is the equation o the graph below? ( x )

34 What is the equation o the graph below? x x 1

35 x What is the equation o the graph below? ( x )

36 Let s generalize the transormations x y ( 1) a( x h ) k Relection across x-axis x y vertical stretch actor ( 1) a( x Translates let/right h) translating up or down k x y ( 1) a x h k x y ( 1) a x h k