CCGPS UNIT 3 Semester 1 COORDINATE ALGEBRA Page 1 of 30. Linear and Exponential Functions

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1 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge of 30 Liner nd Exponentil Functions Nme: Dte: Represent nd solve equtions nd inequlities grphiclly MCC92.A.REI.0 Understnd tht the grph of n eqution in two vribles is the set of ll its solutions plotted in the coordinte plne, often forming curve (which could be line). (Focus on liner nd exponentil equtions nd be ble to dpt nd pply tht lerning to other types of equtions in future courses.) MCC9 2.A.REI. Explin why the x coordintes of the points where the grphs of the equtions y = f(x) nd y = g(x) intersect re the solutions of the eqution f(x) = g(x); find the solutions pproximtely, e.g., using technology to grph the functions, mke tbles of vlues, or find successive pproximtions. Include cses where f(x) nd/or g(x) re liner, polynomil, rtionl, bsolute vlue, exponentil, nd logrithmic functions. Understnd the concept of function nd use function nottion MCC9 2.F.IF. Understnd tht function from one set (clled the domin) to nother set (clled the rnge) ssigns to ech element of the domin exctly one element of the rnge. If f is function nd x is n element of its domin, then f(x) denotes the output of f corresponding to the input x. The grph of f is the grph of the eqution y = f(x). (Drw exmples from liner nd exponentil functions.) MCC9 2.F.IF.2 Use function nottion, evlute functions for inputs in their domins, nd interpret sttements tht use function nottion in terms of context. (Drw exmples from liner nd exponentil functions.) MCC9 2.F.IF.3 Recognize tht sequences re functions, sometimes defined recursively, whose domin is subset of the integers. (Drw connection to F.BF.2, which requires students to write rithmetic nd geometric sequences.) Interpret functions tht rise in pplictions in terms of the context MCC9 2.F.IF.4 For function tht models reltionship between two quntities, interpret key fetures of grphs nd tbles in terms of the quntities, nd sketch grphs showing key fetures given verbl description of the reltionship. Key fetures include: intercepts; intervls where the function is incresing, decresing, positive, or negtive; reltive mximums nd minimums; symmetries; end behvior; nd periodicity. (Focus on liner nd exponentil functions.) MCC9 2.F.IF.5 Relte the domin of function to its grph nd, where pplicble, to the quntittive reltionship it describes. (Focus on liner nd exponentil functions.) MCC9 2.F.IF.6 Clculte nd interpret the verge rte of chnge of function (presented symboliclly or s tble) over specified intervl. Estimte the rte of chnge from grph. (Focus on liner functions nd intervls for exponentil functions whose domin is subset of the integers.) Anlyze functions using different representtions MCC9 2.F.IF.7 Grph functions expressed symboliclly nd show key fetures of the grph, by hnd in simple cses nd using technology for more complicted cses. (Focus on liner nd exponentil functions. Include comprisons of two functions presented lgebriclly.) MCC9 2.F.IF.7 Grph liner nd qudrtic functions nd show intercepts, mxim, nd minim. MCC9 2.F.IF.7e Grph exponentil nd logrithmic functions, showing intercepts nd end behvior, nd trigonometric functions, showing period, midline, nd mplitude. MCC9 2.F.IF.9 Compre properties of two functions ech represented in different wy (lgebriclly, grphiclly, numericlly in tbles, or by verbl descriptions). (Focus on liner nd exponentil functions. Include comprisons of two functions presented lgebriclly.) Build function tht models reltionship between two quntities MCC9 2.F.BF. Write function tht describes reltionship between two quntities. (Limit to liner nd exponentil functions.) MCC9 2.F.BF. Determine n explicit expression, recursive process, or steps for clcultion from context. (Limit to liner nd exponentil functions.) MCC9 2.F.BF.b Combine stndrd function types using rithmetic opertions. (Limit to liner nd exponentil functions.) MCC9 2.F.BF.2 Write rithmetic nd geometric sequences both recursively nd with n explicit formul, use them to model situtions, nd trnslte between the two forms. Build new functions from existing functions MCC9 2.F.BF.3 Identify the effect on the grph of replcing f(x) by f(x) + k, k f(x), f(kx), nd f(x + k) for specific vlues of k (both positive nd negtive); find the vlue of k given the grphs. Experiment with cses nd illustrte n explntion of the effects on the grph using technology. Include recognizing even nd odd functions from their grphs nd lgebric expressions for them. (Focus on verticl trnsltions of grphs of liner nd exponentil functions. Relte the verticl trnsltion of liner function to its y intercept.) Construct nd compre liner, qudrtic, nd exponentil models nd solve problems MCC9 2.F.LE. Distinguish between situtions tht cn be modeled with liner functions nd with exponentil functions. MCC9 2.F.LE. Prove tht liner functions grow by equl differences over equl intervls nd tht exponentil functions grow by equl fctors over equl intervls. MCC9 2.F.LE.b Recognize situtions in which one quntity chnges t constnt rte per unit intervl reltive to nother. MCC9 2.F.LE.c Recognize situtions in which quntity grows or decys by constnt percent rte per unit intervl reltive to nother. MCC9 2.F.LE.2 Construct liner nd exponentil functions, including rithmetic nd geometric sequences, given grph, description of reltionship, or two input output pirs (include reding these from tble). MCC9 2.F.LE.3 Observe using grphs nd tbles tht quntity incresing exponentilly eventully exceeds quntity incresing linerly, qudrticlly, or (more generlly) s polynomil function. Interpret expressions for functions in terms of the sitution they model MCC9 2.F.LE.5 Interpret the prmeters in liner or exponentil function in terms of context. (Limit exponentil functions to those of the form f(x) = bx + k.)

2 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 2 of 30 LESSON 3.0 ALGEBRA EXERCISES Plot the eqution. y 2x 3 2. y x y x 3 3 x 4. 2x 3y 6 5. y 5 6. y 3 4 Wht is the solution for x nd y: X= Y= X= Y= X= Y=

3 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 3 of 30 Solve for x nd y y 3x 9. y x 5 0. y y x 5 Plot both grphs nd determine the solution for x nd y. Use grphing clcultor. y 3x y x. 2. y 2 x y 3 x y x 2 3. y 2 x 4. y y x 2 x

4 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 4 of 30 Lesson 3. Vocbulry A function consists of: Represent Functions s Rules nd Tbles A set clled the domin contining numbers clled inputs nd set clled the rnge contining numbers clled outputs. The input is clled the independent vrible. The output is clled the dependent vrible. The output (dependent vrible) is dependent on the vlue of the input vrible. Exmple: Identify the domin nd rnge of function The input-output tble represents the price of vrious lobsters t fish mrket. Identify the domin nd rnge of the function. Input (pounds) Output (dollrs) $7.60 $0.90 $6.0 $20.50 $25.70 $26.90 Solution: The domin is the set of inputs:.4, 2.2, 3.2, 4.3, 5., 5.3. The rnge is the set of outputs: 7.60, 0.90, 6.0, 20.50, 25.70, PROBLEMS Identify the domin nd rnge of the function.. Input Output Input Output Input Output 3 2 8

5 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 5 of 30 Exmple 2 Mke tble for function The domin of the function y = x + 2 is 0, 2, 5, 6. Mke tble for the function then identify the rnge of the function. x y=x+2 0+2=2 2+2=4 5+2=7 6+2=8 The rnge of the function is 2, 4, 7, 8. PROBLEMS Mke tble for the function. Identify the rnge of the function.. y = 2x - Domin = 0,, 3, 5 x The rnge of the function is: y=2x- 2. y = -4x + 3 Domin = -2, 2, 4, 5 x The rnge of the function is: y=-4x+3 3. y = 0.5x 3 Domin = 0,, 2, 3 x The rnge of the function is: y=0.5x-3 4. y = ½ x 2 Domin = 2, 4, 8, 0 x The rnge of the function is: y=½ x 2

6 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 6 of 30 Exmple 3 Write rule for the function Input Output Solution Let x be the input nd y be the output. Relize tht the output is 3 times the input. Therefore, rule for the function is y = 3x. PROBLEMS Write rule for the function. Input Output Input Output Input Output Input Output

7 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 7 of 30 Lesson 3.2 Exmple Represent Functions s Grphs Grph function Grph the function y=2x with domin 0,, 2, 3, nd 4. Solution Step : Mke n input-output tble. X y Step 2: Plot point for ech ordered pir (x,y). PROBLEMS Grph the function. y 3x Domin: 0,, 2, 3 2. x y x Domin: 0, 2, 4, 6 3. y 2 2 Domin: 0,, 2, 3 2 X y X y X y

8 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 8 of 30 Lesson 3.3 Vocbulry Grph Using Intercepts The x-coordinte of point where the grph crosses the x-xis is the x-intercept. The y-coordinte of point where the grph crosses the y-xis is the y-intercept. Exmple Find the intercepts of grph of n eqution. Find the x-intercept nd the y-intercept of the grph 4x 8y = 24 Solution: To find the x intercept, replce y with 0 nd solve for x 4x 8(0) = 24 Solve for x = 6 To find the y intercept, replce x with 0 nd solve for y 4(0) 8y = 24 Solve for y = -3 Therefore, the x-intercept is 6. The y-intercept is -3. PROBLEMS Find the x-intercept nd the y-intercept of the grph. 5x 7y x 2y 6 3. y x 4 5

9 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 9 of 30 Exmple 2 Use grph to find the intercepts Identify the x-intercept nd the y-intercept of the grph Solution: To find the x-intercept, identify the point where the line crosses the x-xis. The x-intercept is -3. To find the y-intercept, identify the point where the line crosses the y-xis. The y-intercept is -4. PROBLEMS: Identify the x-intercept nd the y-intercept of the grph

10 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 0 of 30 Exmple 3 Use intercepts to grph n eqution Grph 2x + 3y = 6. Lbel the points where the line crosses the xes. Solution: Step : Find the intercepts: To find the y-intercept, set x = 0 nd solve for y. 2(0) + 3y = 6 Solve for y = 2 To find the x-intercept, set y = 0 nd solve for x. 2x + 3(0) = 6 Solve for x =3 Step 2: Plot the points (intercepts) on the corresponding xes. Step 3: Connect the points by drwing line through them.. Grph -3x + 2y = 6. Lbel the points where the line crosses the xes. 2. Grph 5x - 4y = 20. Lbel the points where the line crosses the xes.

11 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge of 30 Lesson 3.4 Find the Slope nd Rte of Chnge Vocbulry The slope of line is the rtio of the verticl chnge (rise) to the horizontl chnge (run) between ny two points of the line. SLOPE = RISE/RUN Exmple Find positive slope Find the slope of the line shown. Solution Pick two convenient points on the line (tht fll on lttice point). Count the boxes (units) you move up. Count the boxes (units) you move to the right. Divide the number of boxes (units) you moved up by the number of boxes (units) you moved to the right to get the slope. Pick the points (,2) nd (-3,-5). You strt t the lower point nd move 7 boxes (units) up. Then you move 4 Boxes (units) to the right. The slope is 7 4. PROBLEMS: Find the slope of the line shown. SLOPE = RISE/RUN

12 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 2 of 30 Exmple 2 Find negtive slope Find the slope of the line shown. Solution Pick two convenient points on the line (tht fll on lttice point). Count the boxes (units) you move up. Count the boxes (units) you move to the left. Divide the number of boxes (units) you moved up by the number of boxes (units) you moved to the left to get the slope. Pick the points (,3) nd (-2,4). You strt t the lower point nd move box (units) up. Then you move 3 Boxes (units) to the left. The slope is 3. PROBLEMS: Find the slope of the line shown

13 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 3 of 30 Lesson 3.5 Grph Using Slope-Intercept Vocbulry A liner eqution (stright line relting x nd y) of the form y = mx + b is written in slope-intercept form. The letter m stnds for the slope (how steep the line is) nd the letter b for the y-intercept (where the line crosses the y-xis). Two prllel lines will not intersect ech other nd hve the sme slope. Two lines re perpendiculr to ech other if they intersect ech other t 90. Their slopes re negtive reciprocls. For exmple if line hs slope of 4 thn line perpendiculr to tht line will hve slope of -¼ (which is the negtive reciprocl of 4). Exmple Identify the slope nd the y-intercept. y x 4 5 Solution: The eqution is in the form y = mx + b. The slope is ¼. The y-intercept is 5. b. 5x 6y 2 Solution: The eqution is not in the form y = mx + b. Rewrite the eqution in slope-intercept form by solving for y. 5x 6y 2 6 y 5x 2 Add 5x to ech side 5 y x 2 Divide ech sideby 6 6 The slope is 5/6. The y-intercept is 2. PROBLEMS: Find the slope nd y-intercept of the line shown.. y = 5x 6 2. y = ½ x y = -6x y = -8x 5. 2x + 3y = 6 6. y = 7 + 3x

14 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 4 of 30 Exmple 2: Grph the eqution 2x + y = 3 using slope intercept form. Solution: Step : Rewrite the eqution in slope-intercept form: y= -2x + 3 Step 2: Identify the slope nd the y-intercept: slope: -2 y-intercept: 3 Step 3: Mrk the point (0,3) s y-intercept on the y-xis. 2 Step 4: From the y-intercept drw slope of cn be rewritten s 2. Therefore, you rise 2 nd run -. Tht mens from the y-intercept of 3 you go up 2 (in the positive y-direction) nd then to the left (in the negtive x-direction). y x PROBLEMS: Grph n eqution using slope intercept form. y = 5x 6 2. y = ½ x y + 6x = 4

15 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 5 of 30 Lesson 3.6 Predict with Liner Models Vocbulry A line tht best represents trend in dt or points is clled the best-fitting line. Using line or its eqution to pproximte vlues between two known points is clled liner interpoltion. Using line or its eqution to pproximte vlues outside the rnge of two known points is clled liner extrpoltion. Exmple Find the eqution of the best fitting line. You re given following dt: X y Find the eqution of the best-fitting line for the dt. Solution: Step : Drw stright line representing the points. Step 2: Find the eqution of the line. The y-intercept is 2 nd the slope is -3. Therefore, the eqution of the line is y = -3x + 2

16 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 6 of 30 PROBLEMS: Find the eqution of the best fitting line Mke sctter plot of the dt. Then find the eqution of the best fitting line. x y Mke sctter plot of the dt. Then find the eqution of the best fitting line. x y

17 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 7 of 30 Lesson 3.7 Compring Liner nd Exponentil Functions Vocbulry A liner function cn be represented in stright line in the form of y mx b. x A exponentil function cn be represented in curved line nd hs the form y ( b ). Exmple Solution John deposits $300 into n ccount which erns 8% interest ech yer on the originl deposit. Jne deposits $300 into n ccount which erns 8% interest ech yer on the yer s end blnce. How much money is in ech ccount fter 3 yers? 8 Firstly, 2% hs to be converted to deciml: 8% John s interest for ech yer is: $ $24 Therefore, the tble below shows the blnce of John s ccount fter ech of the 3 yers: Jck s Account Blnce Yer Yer 2 Yer 3 $324 $348 $372 Jne s interest for yer is just the sme s John s: $ $24. Therefore, Jne will hve blnce of $324 on her ccount. However, for yer 2, the interest is computed using the blnce of $324: $ $26. The interest of $26 is dded to the blnce of $324 to yield n ccount blnce of $350 (324+26=350). Therefore, for yer 3, the interest is computed using the blnce of $350 is: $ $28. Adding the interest of $28 to the ccount blnce of $350 yields $378. The tble below shows the blnce of Jne s ccount fter ech of the 3 yers: Jne s Account Blnce Yer Yer 2 Yer 3 $324 $350 $378 While John gets n identicl interest pyment of $24 per yer Jne s interest pyment increses ech yer since she erns interest on the interest ccumulted. John s interest cn be explined s liner function (sme interest pyment ech yer) while Jne s interest cn be modeled with n exponentil function (interest pyment increses ech yer).

18 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 8 of 30 PROBLEMS. Abdul deposits $500 into n ccount which erns 2% interest ech yer on the originl deposit. Mx deposits $500 into n ccount which erns 2% interest ech yer on the yer s end blnce. How much money is in ech ccount fter 3 yers? Show your nswer in tble formt. Abdul s Account Blnce Yer Yer 2 Yer 3 Mx Account Blnce Yer Yer 2 Yer 3 2. Ted deposits $0,000 into n ccount which erns 9% interest ech yer on the originl deposit. Rick deposits $0,000 into n ccount which erns 9% interest ech yer on the yer s end blnce. How much money is in ech ccount fter 3 yers? Show your nswer in tble formt. Ted s Account Blnce Yer Yer 2 Yer 3 Rick s Account Blnce Yer Yer 2 Yer 3 3. Plot the tble for Abdul nd Mx: Yers on the x-xis nd interest erned (ccount blnce initil deposit) on the y- xis. Abdul Mx Which one hs stright line nd which one shows n upwrd curved line? Which one is liner nd which one is exponentil?

19 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 9 of To encourge communiction between prents nd their children nd to prevent children from hving extremely lrge monthly bills due to dditionl minute chrges, two cell phone compnies re offering specil service plns for students. Tlk Fst cellulr phone service chrges $0.0 for ech minute the phone is used. Tlk Esy cellulr phone service chrges bsic monthly fee of $8 plus $0.04 for ech minute the phone is used. Your prents re willing to purchse for you one of the cellulr phone service plns listed bove. However, to help you become fisclly responsible they sk you to use the following questions to nlyze the plns before choosing one.. How much would ech compny chrge per month if you tlked on the phone for 00 minutes in month? How much if you tlked for 200 minutes in month? b. Build tble nd mke grph for Tlk Fst: X (number of minutes) Y (cost in $) c. Write function rule (in the form y=mx+b), where y represents the cost ($) nd x represents the minutes.

20 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 20 of 30 d. Build tble nd mke grph for Tlk Esy: X (number of minutes) Y (cost in $) e. Write function rule (in the form y= mx+b), where y represents the cost ($) nd x represents the minutes. f. Which compny would be better finncil del if you pln to use the phone for 200 minutes month? Explin your resoning. g. Which compny would be better finncil del if you pln to use the phone for 500 minutes month? Explin your resoning.

21 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 2 of Compre the grphs below.. Which one is shown s stright line nd which one is curved? f( x ) ( 0.5) x g( x) x 3 2 b. Show the grphs in tble formt. g( x) x 3 2 f( x ) ( 0.5) x c. As the vlues of x increse by wht hppens to the corresponding y-vlues of gx ( )? d. As the vlues of x increse by wht hppens to the corresponding y-vlues of f( x )? e. Which function increses in sme increments? Tht function is clled liner function. f. Which function increses in growing increments? Tht function is clled exponentil function. g. Write down n eqution for liner function (different from bove) nd plot the function. h. Write down n eqution for n exponentil function (different from bove) nd plot the function.

22 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 22 of 30 Lesson 3.8 Explicit nd Recursive Formuls Vocbulry An explicit formul llows direct computtion of ny term for sequence, 2, 3, 4,... n,... n ( n ) d where is the first term, nd d is the difference between subsequent terms. Exmple: 2, 2 5, 3 8, 4,... Therefore, 2, nd d 3. To find ny term in the sequence plug the vlues for nd d into the formul: n ( n ) d 2 ( n ) 3 n For sequence, 2, 3, 4,... n,... recursive formul is formul tht requires the computtion of ll previous terms in order to find the vlue of n. Exmple: n n (3) 5 2 n () n (27) n 4 3 etc. Exmple Write the terms of sequence. Write the first five terms of. n 4n 3 b. n ( ) n Solution: Becuse no domin is specified, strt with n=.. b () 3 7 4(2) 3 4(3) 3 5 4(4) 3 9 4(5) ( ) 2 ( ) 3 ( ) 4 ( ) 5 ( )

23 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 23 of 30 PROBLEMS Write the first six terms of the sequence.. n n 3 2. n 4n n n 4. n ( ) 2n n Exmple 2 Solution: Describe the pttern, write the next 3 terms, nd stte rule for the nth term of the sequence 2, 6 2, 20, The difference between 2 nd 6 is 4, the difference between 6 nd 2 is 6, nd the difference between 2 nd 20 is 8. Therefore, the next difference is likely to be 0, then 2, then 4, etc. Accordingly the next three numbers in the sequence should be 30, 42, nd 56. PROBLEMS Describe the pttern, write the next 3 terms, nd stte rule for the nth term of the sequence. 5. 3,6,9,2,... To stte rule in mthemticl terms you hve to relize tht the terms 2, 6, 2, 20 cn be written s (2), 2(3), 3(4), 4(5). The next term should be 5(6) which is 30. Therefore rule for the nth term is n n( n ). 6. 5, 4, 3, 2,...

24 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 24 of 30 7.,,,, ,,,,

25 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 25 of 30 Lesson 3.9 Arithmetic Sequences nd Series Gol: Study rithmetic sequences nd series. In n rithmetic sequence, the difference of consecutive terms is constnt. Tht mens the difference between ny two consecutive terms remins the sme. This difference is clled common difference nd is denoted by d in the formul below. The nth term of n rithmetic sequence with first term nd common difference d is given by: n ( n ) d The expression to dd the terms in n rithmetic sequence is clled n rithmetic series. The sum of n terms is n given by: Sn n 2 Exmple Identify rithmetic sequences. Which ones of the following sequences is rithmetic?. 2,4,6,8,... Solution: Yes, becuse the difference between ll terms is 2. b.,4,7,9,... Solution: No, becuse the difference between ll terms is not the sme. c. 5,9,3,7,... Solution: Yes, becuse the difference between ll terms is 4. PROBLEMS Identify rithmetic sequences. Which ones of the following sequences re rithmetic?. 3,6,9,2, ,8,2,6, ,6,4,2, 2,... 4.,3,5,7,,3,...

26 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 26 of 30 Exmple 2 Write rule for the nth term of the sequence. Then find. 5 PROBLEMS Write rule for the nth term of the sequence. Then find ,7,9,,3,... 9,23,27,3,35,... Solution: The difference between ech term is 4.Therefore d 4. The first term is 9. Therefore, 9. Plug the vlues into the formul n ( n ) d 9 ( n ) 4. Therefore, the rule for the nth term is: n 9 ( n ) 4. To find the 5 th term or 5 replce n with 5: 5 9 (5 ) ,,6,2, ,3,2,, ,2,8,4,20, , 29, 33, 37,...

27 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 27 of 30 Exmple 3 Find the sum of the first 25 terms of the rithmetic series9,23,27,3,35,.... Solution: The difference between ech term is 4.Therefore d 4. The first term is 9. Therefore, 9. Plug the vlues into the formul n ( n ) d 9 ( n ) 4. Therefore, the rule for the nth term is: n 9 ( n ) 4. To find the 25 th term or 25 replce n with 25: 25 9 (25 ) Plug the vlues for, 25, d into the formul S n n n or = S PROBLEMS Find the sum of the first n terms of the rithmetic series. 0. 5,7,9,,3,...; n 9. 25,35,45,55,...; n ,3,2,,...; n ,8,,4,...; n 0

28 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 28 of 30 Exmple: Find n explicit nd recursive formul for the sequence 5,8,,4,7,... Solution: An explicit formul cn be written in the form n ( n ) d where is the first term, nd d is the difference between subsequent terms. The first term, 5. The difference between ech term, d 3. To find ny term in the sequence plug the vlues for nd d into the formul: n ( n ) d 5 ( n ) 3 n To check whether the explicit formul for the sequence is correct check by putting in the vlues for nd d for let s sy the 4 th term: 4 ( n ) d 5 (4 )3 5 (3)3 4 This is indeed correct. A recursive formul is formul tht requires the computtion of ll previous terms in order to find the vlue of n. In our cse 5. The next term cn be written s n n 3. To check whether this is correct put in the vlues for the first 4 terms: n n n 4 3 etc.

29 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 29 of 30 SEQUENCES ARE FUNCTIONS 4. Represent the function f ( n) 2n in tble form nd grphiclly. Fill in the remining points. f ( n) 2n Write n explicit formul for the function in the form n ( n ) d where is the first term nd d is the difference between two consecutive terms. 6. Write recursive formul for the function. (Exmple : the previous term). n 3 n. The next term is expressed in terms of 5

30 CCGPS UNIT 3 Semester COORDINATE ALGEBRA Pge 30 of Represent the function f ( n) n 3 in tble form nd grphiclly. f ( n) n Write n explicit formul for the function in the form n ( n ) d where is the first term nd d is the difference between two consecutive terms. 9. Write recursive formul for the function. (Exmple : the previous term). n 3 n The next term is expressed in terms of 5