EQUATIONS REVIEW I Lesson Notes. Example 1. Example 2. Equations Review. 5 2 x = 1 6. Simple Equations

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1 Equaions Review x + 3 = 6 EQUATIONS REVIEW I Example Simple Equaions a) a - 7 = b) m - 9 = -7 c) 6r = 4 d) 7 = -9x Example Simple Equaions a) 6p + = 4 b) 4 = 3k + 6 c) 9 + k = + 3k d) 8-3n = -8n + 3

2 EQUATIONS REVIEW I Equaions Review x + 3 = 6 Example 3 Simple Equaions a) -(a - ) = 3a + 7 b) x - (3x + ) = 8 c) - x = 4 + (x - ) d) 3x - (x - ) = -(3x - 9) Example 4 Cross Muliplicaion 6 a) b) c) d) x = -3 3 = x x = 9 4 x = -

3 Equaions Review x + 3 = 6 EQUATIONS REVIEW I Example Cross Muliplicaion a) 3 x = -3 b) - x = 3 c) x + = 4 d) 3 (x - 4) = 6

4 EQUATIONS REVIEW I Equaions Review x + 3 = 6 Example 6 Cross Muliplicaion a) - = -9 - a 4 - (-3) b) - = a - (-3)

5 Equaions Review x + 3 = 6 EQUATIONS REVIEW I c) (-9) = - (-) a - (-4) d) -7 - a = 6 - (-3) -

6 EQUATIONS REVIEW II Equaions Review x + 3 = 6 Example 7 Lowes Common Muliple (LCM) a) 3 + x 7 = 3 b) x + 3 = 6

7 Equaions Review x + 3 = 6 EQUATIONS REVIEW II c) 3 4 x 3 - = d) - = x

8 EQUATIONS REVIEW II Equaions Review x + 3 = 6 Example 8 Lowes Common Muliple (LCM) a) 3 4 x + 3 x - 6 = b) x - - x + 7 =

9 Equaions Review x + 3 = 6 EQUATIONS REVIEW II c) x = x d) x + = - 3 (x + 8) 4

10 EQUATIONS REVIEW II Equaions Review x + 3 = 6 Example 9 Cross Muliply or LCM? a) x 3 = 4 b) 3 x - = 4

11 Equaions Review x + 3 = 6 EQUATIONS REVIEW II c) 3 (x - ) = x - d) x = x

12 EQUATIONS REVIEW II Equaions Review x + 3 = 6 Example Isolae y a) 4x - y + 9 = b) y + = (x - )

13 Equaions Review x + 3 = 6 EQUATIONS REVIEW II c) y - 7 = - (x + 4) 9 d) 3 4 x - 3 y - 6 =

14 EQUATIONS REVIEW II Equaions Review x + 3 = 6 Example General Form: Ax + By + C = a) y = - 3 x + 3 b) y - = (x + ) 4 c) y + = - (x - 3) 7

15 Equaions Review x + 3 = 6 EQUATIONS REVIEW II Example Plugging in Numbers a) y = - (x = 8, y =?) 4 x + 3 b) y + 3 = - (x - ) (x =?, y = -) c) 3 4 x - 3 y - 6 = (x =?, y = 8)

16 d() Relaions and Funcions LESSON FIVE - Inerpreing Graphs Inroducion In a m fish race, here are hree compeiors. Teleporing Fish - has he abiliy o insanly warp from locaion o locaion. Insan-Speed Fish - can reach any desired speed insanly wihou acceleraing. Real-World Fish - mus speed up and slow down, jus like objecs in realiy. a) Teleporing Fish spends he firs s of he race resing a he sar line. He hen warps o he midpoin of he rack and ress for anoher seconds. Finally, he warps o he end and wais seconds while he oher fish arrive. Graph his moion. d() b) Insan-Speed Fish begins he race a. m/s, and susains ha speed for seconds unil she reaches he midpoin. Afer resing for seconds, she resumes her speed of. m/s and heads o he finish line. d() c) Real-World Fish acceleraes o a speed of. m/s in 6 seconds, holds ha speed for 8 seconds, and hen deceleraes o zero in 6 seconds - his brings him o he midpoin. Afer resing for seconds, Real-World fish repeas he moion - accelerae for 6 seconds, hold he speed for 8 seconds, and decelerae for 6 seconds. This brings him o he finish line. d() 3 4 6

17 Relaions and Funcions LESSON FIVE - Inerpreing Graphs d() Example Alex walked halfway o school, bu realized he forgo his calculaor. He urned around, ran back home, and searched his room for five minues rying o find he calculaor. He hen ran wo-hirds of he way back o school, bu go ired and had o walk he remaining hird. Draw a graph represening Alex's journey. Assume insan speed changes. disance Drawing he graph exacly requires calculaions using ime =. speed Disance from home o school Alex's running speed Alex's walking speed Find ordered pairs ha will le you draw he graph. Use he space below for your work. 6 m m/s m/s i) walking o school ii) running back home iii) looking for calculaor iv) running o school v) walking o school d()

18 d() Relaions and Funcions LESSON FIVE - Inerpreing Graphs Example Each of he following graphs represens a poenial pah Naomi can ake from home o school. Deermine if each graph represens a possible or impossible moion. a) d() b) d() c) d() Possible: Yes No Possible: Yes No Possible: Yes No Example 3 Represen each of he following moions in graphical form. a) A ball is hrown sraigh up and falls back down. b) A rubber ball is dropped and bounces hree imes. c) The swimming pool below is filled wih waer. h() h() h()

19 Relaions and Funcions LESSON FIVE - Inerpreing Graphs d() Example 4 The following able shows he Canada Pos price lis for mailing leers wihin Canada. Leer Mass up o (and including) 3 g up o (and including) g up o (and including) g up o (and including) g up o (and including) 3 g up o (and including) 4 g up o (and including) g Price $.7 $. $. $. $.7 $3. $3. a) Graph his daa b) Sae he domain and range C(m) 4. Domain: Range: m

20 Graph y = x x - - y 4 4 Relaions and Funcions LESSON ONE - Graphing Relaions Inroducion a) Wrie a senence ha describes his relaion. Cailin rides her bike o school every day. The able of values below shows her disance from home as ime passes. ime (minues) disance (meres) b) Represen his relaion wih ordered pairs. 3 7 c) Represen his relaion wih an arrow diagram. 4 d) Wrie an equaion for his scenario. d 7 e) Graph he relaion. 3 4 Example a) y = -x + 3 For each relaion, complee he able of values and draw he graph. b) y = x x y x y

21 Relaions and Funcions LESSON ONE - Graphing Relaions Graph y = x x - - y 4 4 Example a) y = x For each relaion, complee he able of values and draw he graph. Sae if he relaion is linear or non-linear. b) y = x + x y x y Example 3 For each scenario, sae he dependen variable, he independen variable, and he rae. Wrie he equaion. a) A frui vendor generaes a revenue of R dollars by selling n boxes of plums a $3 each. i) he dependen variable is. ii) he independen variable is. iii) he rae is. iv) he equaion is. b) A runner wih a speed of 9 m/s can run d meres in seconds. i) he dependen variable is. ii) he independen variable is. iii) he rae is. iv) he equaion is. c) A diver experiences a pressure of P kilopascals a a deph of d meres. Underwaer pressure increases a kilopascals/mere. i) he dependen variable is. ii) he independen variable is. iii) he rae is. iv) he equaion is.

22 Graph y = x x - - y 4 4 Relaions and Funcions LESSON ONE - Graphing Relaions Example 4 a) Wrie an equaion for his scenario. b) Generae a able of values. Tickes o a concer cos $ each. The revenue from icke sales is R, and he number of ickes sold is n. c) Draw he graph. n R R d) Is he relaion 6 coninuous or discree? ADMIT ONE TICKET Oc. 6 8: PM ADMIT ONE 3 4 n Example a) Wrie an equaion for his scenario. A cylindrical ank is being filled wih waer a a rae of 3 L/min. The volume of waer in he ank is V, and he elapsed ime is. b) Generae a able of values. V c) Draw he graph. V d) Is he relaion coninuous or discree? 3 4 Example 6 A relaion is represened by 4x + y = 8. a) Isolae y so his relaion can be graphed. b) Generae a able of values. x y c) Draw he graph. d) Is he relaion coninuous or discree?

23 Relaions and Funcions LESSON ONE - Graphing Relaions Graph y = x x - - y 4 4 Example 7 Nick, a salesman, earns a base salary of $6/week plus an 8% commission on sales. The amoun of money Nick earns in a week is E, and he oal value of his sales is s. a) Wrie an equaion ha relaes he variables. b) Complee he able of values. s E 3 4 g) If Nick makes $6 in sales one week, wha will his earnings be? c) Draw he graph. E h) How much will Nick have o sell if he makes $6 in one week? s d) Is his relaion linear or non-linear? e) Is his relaion discree or coninuous? f) Wha are he dependen and independen variables?

24 Domain { x -6 < x 3, x ε R } Range { y - y <, y ε R } Relaions and Funcions LESSON TWO - Domain and Range Inroducion a) Wrie he domain and range of his graph in senence form. b) Wrie he domain and range of his graph as number lines. Domain: Range: Domain: Range: c) Wrie he domain and range of his graph in se noaion. d) Wrie he domain and range of his graph as a discree lis. Domain: Range: Domain: Range:

25 Relaions and Funcions LESSON TWO - Domain and Range Domain {x -6 < x 3, x ε R} Range {y - y <, y ε R} e) Wrie he domain and range of his graph using inerval noaion. Domain: Range: Example Wrie he domain of each number line. a) Domain: b) Domain: c) Domain: d) Domain: e) Domain:

26 Domain {x -6 < x 3, x ε R} Range {y - y <, y ε R} Relaions and Funcions LESSON TWO - Domain and Range Example domain and range of discree graphs. a) Domain: b) Domain: Range: Range: Example 3 domain and range of coninuous graphs. a) Domain: b) Domain: Range: Range: Example 4 domain and range of graphs wih endpoins a) Domain: b) Domain: Range: Range:

27 Relaions and Funcions LESSON TWO - Domain and Range Domain {x -6 < x 3, x ε R} Range {y - y <, y ε R} Example domain and range of parabolas and enclosed shapes a) Domain: b) Domain: Range: Range: Example 6 A Ferris wheel has a radius of m and makes one complee revoluion every wo minues. Riders board he wheel a a heigh of one mere above he ground. A ride lass for hree revoluions of he wheel. The graph of he moion is shown below. Sae he domain and range, in as many ways as possible. Se Noaion Senence Discree Lis Number Lines h Inerval Noaion 3 4 6

28 x = y - inercep y = x - inercep (, y) (x, ) Relaions and Funcions LESSON FOUR - Inerceps Inroducion Find he inerceps and draw he graph. a) y = 4x - 8 b) f(x) = 3 x + c) d() = d() Example a) The funcion f(x) = x + k has a y-inercep of -. Find he value of k. b) The funcion f(x) = 3x + k has an x-inercep of -. Find he value of k.

29 Relaions and Funcions LESSON FOUR - Inerceps x = y - inercep y = x - inercep (, y) (x, ) Example A cylindrical ank wih 4 L of waer is being drained a a rae of L/min. a) Graph he volume of he ank. V() b) Wrie a funcion o represen his scenario. c) Wha does each inercep represen? d) Sae he domain and range.

30 x = y - inercep y = x - inercep (, y) (x, ) Relaions and Funcions LESSON FOUR - Inerceps Example 3 A mounain climber is a he peak of a mounain wih an aliude of 4 m. I akes 8 hours for he climber o reurn o ground level. The climber can descend he mounain a an average speed of 7 m/hour. a) Graph he heigh of he mounain climber. h() b) Wrie a funcion o represen his scenario. c) Wha does each inercep represen? d) Sae he domain and range.

31 f(x) Relaions and Funcions LESSON THREE - Funcions Inroducion For each of he following funcions, complee he able of values and draw he graph. a) f(x) = x + 4 x f(x) - - b) f(x) = 3x 4 x f(x) - - c) f(x) = x - 3 x f(x) - -

32 Relaions and Funcions LESSON THREE - Funcions f(x) Example For each funcion, calculae f(3). a) d) b) e) 3 c) f) 3 Example Use he graph of each funcion o deermine he value of f(3). a) c) b) d)

33 f(x) Relaions and Funcions LESSON THREE - Funcions Example 3 Deermine which of he following graphs represens a funcion. a) b) c) d) Funcion: Yes No Funcion: Yes No Funcion: Yes No Funcion: Yes No Example 4 a) Given f(x) = x +, he poin (k, ) exiss on he graph. Find k. b) Given f(x) =, he poin (k, -3) exiss on he graph. Find k. c) Does he poin (-, 8) exis on he graph of f(x) = -7x + 3?

34 Relaions and Funcions LESSON THREE - Funcions f(x) Example A speed walker walks wih a speed of 6 km/hour. a) Use a able of values o deermine he disance walked in he firs five hours. 3 4 d d) Sae he dependen and independen variables. dependen: independen: b) Wrie he disance funcion. Disance Funcion e) Wrie he domain and range. Domain: Range: c) Draw he graph of his funcion. Is he graph coninuous or discree? d() 3 f) How far does he speed walker ravel in.4 hours? g) How long does i ake for he speed walker o walk.6 km? 3 4

35 f(x) Relaions and Funcions LESSON THREE - Funcions Example 6 The cos of a sandwich is $4.4 wih wo oppings, and $. wih five oppings. a) Use a able of values o deermine he cos of he sandwich for he firs five oppings. n 3 4 C d) Sae he dependen and independen variables. dependen: independen: e) Wrie he domain and range. b) Wrie he cos funcion. Cos Funcion Domain: Range: c) Draw he graph of his funcion. Is he graph coninuous or discree? There are oppings available. C(n) f) Wha is he price of a sandwich wih seven oppings? g) How many oppings are on a $.8 sandwich? n