Math-3 Lesson 1-4. Review: Cube, Cube Root, and Exponential Functions

Transcription

1 Math- Lesson -4 Review: Cube, Cube Root, and Eponential Functions

2 Quiz - Graph (no calculator):. y. y ( ) 4. y

3 What is a power? vocabulary Power: An epression ormed by repeated Multiplication o the same actor. 4 Eponent Coeicient Base The base is used as a actor the eponent number o times. * * * *

4 Cubing Function ( ) What is the domain o the unction? What is the range o the unction? All real numbers. All real numbers.

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

6 What is the transormation o the parent unction? ( ) ( ) ( ) ( ) 4 Relected across -ais. Right, Up 4. Graph the ollowing equation (without a calculator). ( ) ( )

7 Cubed Root (or rd root) Inde Radical 5 Radicand 4-5 7

8 5 rd Root 5 some number equals the rd root o 5. ( ) 5 I I cube both sides, what does the equation become? 5 Does the same value o make both equations true? 4-5 8

9 What shape does it have? ) ( Build a table o values or the nice domain elements: -8, -, 0,, y y y ( ) ( 8)? 8? y is the output value corresponding to the input value -8 y

10 Cubed Root Function ( ) What is the range o the unction? So you can take the rd root o a negative number. All real numbers. What is the domain o the unction? All real numbers.

11 What is the transormation o the parent unction? ( ) ) ( ( ) 4 Relected across -ais. Right, Up 4. Graph the ollowing equation (without a calculator). ( )

12 What shape does it have? ( ) Build a table o values or the nice domain elements: -, -, 0,, y ( )?? Do you remember the negative eponent property? Fill in the rest o the table.

13 Eponential Function ( ) Will the y value ever reach zero (on the let end o the graph)? Why not? As the input becomes more negative, the denominator o the raction becomes bigger so the overall value o the raction gets smaller. The denominator o a raction can never make a raction = 0.

14 Eponential Function ( ) (0) =? = y-intercept g ( ) () What is the transormation o the parent? (0) =? = y-intercept General Form o an any base eponential unction: ( ) ab vertical stretch actor base o the eponential

15 The Initial Value I the input variable was time, the previous unction would look like: ( t) () Since negative time doesn t make sense, what is the domain o this unction? [0, ) ( what input values are allowed?) The initial value occurs when t = 0. t What is the initial value o (t)?? ( 0) 0 (0) =? (0) ()?

16 Vocabulary: The Initial Value ( ) 7() The initial value o the unction is the coeicient o the power. What is he initial value o the ollowing unctions? g ( ) (5) g( ) 7

17 Your turn: What is the initial value o: ( t) 0.5() t What is the initial value o: ( ) ab The y intercept is a point on the y-ais. What is the -value o every y-intercept?

18 () (output values) Eponential Growth (input value) b > a is the initial value (0) = a b is called the growth actor ( ) ab ( ) () Table o values () 0 () 0 () () 6 () () () ().5 () 0.75

19 Eponential Growth ( ) ab ( ) () () (output values) What do we call the line: y = 0? (input value) Horizontal asymptote I we restricted the domain to only whole numbers (,,, ), what would be the range? 4 What kind o sequence is the range? y Geometric Sequence!!!

20 () (output values) Eponential Decay 0 < b < (input value) a is the initial value (0) = a b is called the decay actor ( ) ab ( ) 4(0.5) Table o values () 0 4(0.5) 0 4(0.5) 4 4(0.5) 4(0.5) 4(0.5) (0.5) (0.5) 4(0.5) 8 6 ½ ½ ½ ½

21 Eponential Data: what is the equation? -values increment by one each time. y-values increment by the same actor each time. This number is the growth actor The growth actor is the base o the eponential.

22 Find the unction or this data. g() is eponential g( ) ab a 4 b g ( ) 4()

23 Your turn. Find h(). h is eponential h( ) ab. h(0) = 8 8 ab 0. h() = h( ) 8 4 h( ) 8b a b 4

24 Your turn: ( ) 5(). What is the initial value a?. Where does the graph cross the y-ais? 4. What is the growth actor? 5. Is this unction growth or decay? 6. Answer questions - 5 or: ( )

25 Transorming Eponential Functions Describe how to transorm the graph o: Into the graph o : ( ) ( )

26 Transorming Eponential Functions ( ) ( ) The graph o rom ( ) ( ) ( ) can be obtained by relecting it across the y-ais.

27 Transormations o the Eponential Function ( ) () g( ) () 5 h( ) () 4 k( ) ().5 Horizontal asymptote: y = 4 Horizontal asymptote: y =.5

28 Your turn: 7. Describe how to transorm the graph o: ( ) Into the graph o : ( ) () 4 What is the horizontal asymptote?

29 Eponential Growth and Decay ( ) ab For what range o values o b will result in eponential growth? For what range o values o b will result in eponential decay? b > 0 < b <

30 Parent unction ( ) y b Relection across - ais ( ) a*( b) Vertical stretch actor The Transormation Equation h Horizontal shit k Growth actor Vertical shit And the Horizontal asymptote g( ) 4() ( ) 5

31 vertical shit and horizontal Asymptote ( ) ( ) ab ( )( c) d I negative: Relect across -ais Initial value: Crosses y-ais here Horizontal shit I negative: Relect across y-ais Growth actor: ( ) 4() 4() ( )