Functions: Review of Algebra and Trigonometry

Transcription

1 Sec. and. Functions: Review o Algebra and Trigonoetry A. Functions and Relations DEFN Relation: A set o ordered pairs. (,y) (doain, range) DEFN Function: A correspondence ro one set (the doain) to anther set (the range) such that each eleent in the doain corresponds to eactly one eleent in the range. Eaple: Deterine whether each o the ollowing is an eaple o a unction or not..),, 5,.),,4, 5, 4.) y 0 4.) y Vertical Line Test or Functions Vertical Line Test or Functions: I any vertical line intersects a graph ore than once, then the graph is not a unction. Eaple: Deterine whether each o the ollowing is a unction or not by the Vertical Line Test. Desiré Taylor Math 4

2 B. Doain and Range o a Function DEFN Doain: Input -values (i.e. All o the values o that I ay plug into a unction.) DEFN Range: Output y-values (i.e. All o the values o y that a unction can attain) Function Notation: () = y Eaple: Give the doain and range (in interval notation) or each o the ollowing.) Doain: Doain: Doain: Range: Range: Range: Eaple: Give the doain or each o the ollowing (in interval notation and as an inequality).) g.) ) g 5.) h ln6 * The unctions or which we will ost requently have doain restrictions (in this course) are: ractions (aka rational unctions), radicals and logariths. Desiré Taylor Math 4

3 C. Linear Models Deinition rise Slope = = = run y y y = Slope Slope-Intercept For Point-Slope For y y General For A B y C 0 y (Use when given two points to ind slope) y y b (Use when given slope and y - intercept) (Use when given one point and slope) Horizontal Line Vertical Line y b (where b = constant) c (where c = constant) Parallel Lines Two lines are parallel i and only i they have the sae slope. For two lines y b and y b we have y Perpendicular lines y Two lines are perpendicular i and only i the product o their slope = -. y For two lines y b and y b we have y Eaples: Find the equation o the line:.) that passes through point (0, -) with slope = -.) that passes through points (, -) and (4, 5) Desiré Taylor Math 4

4 .) that passes through point (0, 0) and is parallel to the line y 9 4.) that passes through point (, -4) and is perpendicular to the line y 5.) Find the slope and y-intercept o the line 9 y 0 D. Classes o Functions. Power Functions For any real nuber, a unction in the or is called a Power Function. Polynoials Deinition n n n A polynoial unction is a unction in the or an an an a a0 a 0 0 and n is a positive integer. where Eaples: State whether each is a polynoial:.) g 5 6.) 7.) 8 4.) h 5 5.) 6 4 Desiré Taylor Math 4 4

5 . Rational Functions A Rational Function is the quotient o two polynoial unctions: A Rational Function is a unction o the or ( ) p( ) an q( ) b n a b n n... a... b a 0 b 0 Asyptotes An asyptote is an iaginary line that the graph o a unction approaches as the unction approaches a restricted nuber in the doain or as it approaches ininity. I. Locating Vertical Asyptotes p( ) I ( ) is a rational unction, p() and q () have no coon actors and n is a zero o q (), then the q( ) line n is a vertical asyptote o the graph o (). II. Locating Horizontal Asyptotes n n p( ) an an Let ( ) q( ) b b... a... b a 0 b 0 i. I n <, then y 0 is the horizontal asyptote ( Botto Heavy ) ii. an I n =, then the line y b iii. is the horizontal asyptote ( Equal Degree ) I n >, there is NO horizontal asyptote. (But there will be a slant/oblique asyptote) ( Top Heavy ) Eaples: Find all vertical and horizontal asyptotes:.) 5 ( ).) 5 g ( ).) 7 ( ) 5 h 4.) k ( ) Desiré Taylor Math 4 5

6 4. Trigonoetric Functions sin() cos() tan() csc() sec() cot() 5. Eponential and Logarithic Functions DEFN: An eponential unction is a unction in the or DEFN: A logarithic unction is a unction in the or log y = log b a. (i.e. the variable is in the eponent) a. (i.e. the variable is in the epression) y is equal to log base b o - Here b is the BASE NUMBER and is the VARIABLE. log b = y eans eactly the sae thing as b y = y = y = log () Coparison o the two graphs, showing the inversion line in red. Desiré Taylor Math 4 6

7 E. Transorations o Functions Vertical Shits C Horizontal Shits C C C Moves Graph UP C units Moves Graph DOWN C units Moves Graph RIGHT C units Moves Graph LEFT C units Vertical and Horizontal Relections Vertical Stretching/ Copressing c Flips Graph About -ais or c Flips Graph About y-ais Graph Vertically Stretches by a Factor o C c or 0 c Graph Vertically Shrinks by a Factor o C Eaple: Use the given graph o () to sketch each o the ollowing. () a. () + b. ( + ) c. () d. (-) F. Cobinations o Functions. Piecewise-Deined Functions A Piecewise Function is a unction that has speciic (and dierent) deinitions on speciic intervals o. 0 0 Doain: Range: Desiré Taylor Math 4 7

8 . Sus, Dierences, Products and Quotients o Functions Eaple Su g g Dierence g g Product g g Quotient. I () = + and g() = ind each o the ollowing. g g a. (4) b. g() c. ( 4) d. () + g() e. ()g(). Coposition o Functions Eaple: Notation g g.) For the unctions and g ind a.) g b.) g c.) g d.) g g Desiré Taylor Math 4 8

9 Eaple: For the unctions () and g() given in the graph ind a.) c.) e.) g b.) g g d.) g 4.) g g4 G. Syetry Syetry: Even unctions () = (- ). Syetric about the y-ais. I (a,b) then (-a,b) Odd unctions (- ) = - (). Syetric about the origin. I (a,b) then (-a,-b). State whether the ollowing unctions are even, odd, or neither. a. () = b. () = 4 c. () = d. () = sin() e. () = cos(). () = Desiré Taylor Math 4 9

10 H. Function Properties - Increasing unctions rise ro let to right Decreasing unctions all ro let to right Positive unctions are above the -ais Negative unctions are below the -ais *For all o these above, you use the -values to state your answers!. Find each o the ollowing using the given unction. a. () > 0 b. () 0 c. increasing d. decreasing e. doain and range. Find each o the ollowing using the given unction. a. () > 0 b. () 0 c. increasing d. decreasing e. doain and range Desiré Taylor Math 4 0

11 Sec. Review o Algebra and Trigonoetry A. Functions and Relations DEFN Relation: A set o ordered pairs. (,y) (doain, range) DEFN Function: A correspondence ro one set (the doain) to anther set (the range) such that each eleent in the doain corresponds to eactly one eleent in the range. Eaple: Deterine whether each o the ollowing is an eaple o a unction or not..),, 5,.),,4, 5, 4.) y 0 4.) y Vertical Line Test or Functions Vertical Line Test or Functions: I any vertical line intersects a graph ore than once, then the graph is not a unction. Eaple: Deterine whether each o the ollowing is a unction or not by the Vertical Line Test. Desiré Taylor Math 4

12 B. Doain and Range o a Function DEFN Doain: Input -values (i.e. All o the values o that I ay plug into a unction.) DEFN Range: Output y-values (i.e. All o the values o y that a unction can attain) Eaple: Give the doain and range (in interval notation) or each o the ollowing.) Doain: Doain: Doain: Range: Range: Range:.) g.) ) g 5.) h ln6 * The unctions or which we will ost requently have doain restrictions (in this course) are: ractions, radicals and logariths. Desiré Taylor Math 4

13 C. Linear Models Deinition rise Slope = = = run y y y = Slope Slope-Intercept For Point-Slope For y y General For A B y C 0 y (Use when given two points to ind slope) y y b (Use when given slope and y - intercept) 0 0 (Use when given one point and slope) Horizontal Line Vertical Line y b (where b = constant) c (where c = constant) Parallel Lines Two lines are parallel i and only i they have the sae slope. For two lines y b and y b we have y Perpendicular lines y Two lines are perpendicular i and only i the product o their slope = -. y For two lines y b and y b we have y Eaples: Find the equation o the line:.) that passes through point (0, -) with slope = -.) that passes through points (, -) and (4, 5) Desiré Taylor Math 4

14 .) that passes through point (0, 0) and is parallel to the line y 9 4.) that passes through point (, -4) and is perpendicular to the line y 5.) Find the slope and y-intercept o the line 9 y 0 D. Classes o Functions. Power Functions For any real nuber, a unction in the or is called a Power Function. Polynoials Deinition n n n A polynoial unction is a unction in the or an an an a a0 a 0 0 and n is a positive integer. where Eaples: State whether each is a polynoial:.) 5 6 g.) 7.) h 5 4.) 8 Desiré Taylor Math 4 4

15 . Rational Functions A Rational Function is the quotient o two polynoial unctions: A Rational Function is a unction o the or ( ) p( ) an q( ) b n a b n n... a... b a 0 b 0 Asyptotes An asyptote is an iaginary line that the graph o a unction approaches as the unction approaches a restricted nuber in the doain or as it approaches ininity. I. Locating Vertical Asyptotes p( ) I ( ) is a rational unction, p() and q () have no coon actors and n is a zero o q (), then the q( ) line n is a vertical asyptote o the graph o (). II. Locating Horizontal Asyptotes n n p( ) an an Let ( ) q( ) b b... a... b a 0 b 0 i. I n <, then y 0 is the horizontal asyptote ii. an I n =, then the line y b iii. is the horizontal asyptote I n >, there is NO horizontal asyptote. (But there will be a slant/oblique asyptote.) Eaples: Find all asyptotes:.) 5 ( ).) 5 g ( ).) 7 ( ) 5 h 4.) k ( ) Desiré Taylor Math 4 5

16 4. Trigonoetric Functions sin() cos() tan() csc() sec() cot() 5. Eponential and Logarithic Functions DEFN: An eponential unction is a unction in the or DEFN: A logarithic unction is a unction in the or log y = log b a. (i.e. the variable is in the eponent) a. (i.e. the variable is in the epression) y is equal to log base b o - Here b is the BASE NUMBER and is the VARIABLE. log b = y eans eactly the sae thing as b y = y = y = log () Coparison o the two graphs, showing the inversion line in red. Desiré Taylor Math 4 6

17 E. Transorations o Functions Vertical Shits C Horizontal Shits C C C Moves Graph UP C units Moves Graph DOWN C units Moves Graph RIGHT C units Moves Graph LEFT C units Vertical and Horizontal Relections Vertical Stretching/ Copressing c Flips Graph About -ais or c Flips Graph About y-ais Graph Vertically Stretches by a Factor o C c or 0 c Graph Vertically Shrinks by a Factor o C F. Cobinations o Functions. Piecewise-Deined Functions A Piecewise Function is a unction that has speciic (and dierent) deinitions on speciic intervals o Sus, Dierences, Products and Quotients o Functions Su g g Dierence g g Product g g Quotient g g Desiré Taylor Math 4 7

18 . Coposition o Functions () = +5+ () = () +5 ()+ g g k k h 5 h 5 5 Eaple: Notation g g.) For the unctions and g ind a.) g b.) g c.) g d.) g g Eaple: For the unctions () and g() given in the graph ind a.) c.) e.) g b.) g g d.) g 4.) g g4 Desiré Taylor Math 4 8

19 Sec. The Liit o a Function A. Liits L DEFN: a The it o () as approaches a, equals L. (Where is the unctions value headed as is "on its way" to a?) a a The it o () as approaches a ro the LEFT The it o () as approaches a ro the RIGHT B. Techniques o Solving Liits. Evaluation - When possible (without violating doain rules) "plug it in". Eaple:.).). Factoring/Manipulation (then Evaluation) - Factor epressions and cancel any coon ters. Eaple: 4.) 4 6.). Table - Set up a table as approaches the it ro the let and ro the right. Eaple: sin.) 0 Desiré Taylor Math 4

20 4. Graphing - Graph the unction and inspect. (Warning: Your graphing calculator ight not always indicate a hole or sall discontinuity in a graph. Be sure to always check the doain or restrictions.) Eaple: sin.) 0 More Eaples:.) 0.) a a ga g a.) 7 i 4 i 4 i Desiré Taylor Math 4

21 4.) g g5 g 5 g g 5 g g 5 g C. Average Velocity DEFN: Eaple: Velocity Distance Tie A ball is thrown up straight into the air with an initial velocity o 55 t/sec, its height in eet t seconds is given by y 75t 6t. a.) Find the average velocity or the period beginning when t= and lasting (i) 0. seconds (i.e. the tie period [,.]) (ii) 0.0 seconds (iii) 0.00 seconds b.) Estiate the instantaneous velocity o the ball when t=. Desiré Taylor Math 4

22 Sec.4 Calculating Liits A. Liit Laws.).).) 4.) 5.) 6.) 7.) Assue that and g are unctions and c is a constant. a a a a a a g g a a g g a c c a a g g g a a a g n n c c a a ; a a g 0 8.) 9.) a a a n n a 0.) a n n a.) a n n a Desiré Taylor Math 4

23 Eaple:.) a.) g b.) c.) g d.) g.) a.) b.) c.) d.) e.) Desiré Taylor Math 4

24 B. Calculating Liits Direct Substitution Property: I is a polynoial or rational unction and a Doain, then a a Eaples:.).) ) 4 4.) cos 5.) ) Theore We say that a it eists when the it ro the let equals the it ro the right. Eaples: i i a.) h Find h L L a a Desiré Taylor Math 4

25 Theore Squeeze Theore: I g h and h L Eaples:.) Find cos 0 then g L a when is near a, a a More Eaples:.).) Desiré Taylor Math 4 4

26 Sec.5 Continuity A. Deinition o Continuity DEFN: A unction is continuous at a nuber a i: (i) (ii) a eists eists a (iii) a a A unction is deined as continuous only i it is continuous at every point in the doain o the unction. Eaples: For each, deterine whether the unction is continuous (i.e. Is.).) = A A?).) 4.) Eaples: For each, deterine whether the unction is continuous. I not, where is the discontinuity?.) 0 5.) Desiré Taylor Math 4

27 i 5 i.) h 4.) k (Step unction: i.e. k int ) 5.) 6.) Desiré Taylor Math 4

28 DEFN: A unction is continuous ro the RIGHT at a nuber a i: a a A unction is continuous ro the LEFT at a nuber a i: a a Eaple:.) Is is continuous ro the LEFT or RIGHT at a.) b.).) Show that has a jup discontinuity at 9 by calculating the its ro the let and right at 9. Theore I and g are unctions that are continuous at a nuber a, and c is a constant, then the ollowing are also continuous at a : (i) g (ii) g (iii) g (iv) g (v) c or c g i g a 0 Theore A polynoial unction is continuous everywhere A rational unction is continuous everywhere it is deined Desiré Taylor Math 4

29 Theore Interediate Value Theore I is a unction that is continuous on a closed intervala, b where a b and N is a nuber such that a N b. Then there eist a nuber c such that a c b and c N. Eaples:.) Show that has a root on the interval,.) Let be a continuous unction such that 0 9. Then the Interediate Value Theore iplies that 0 on the interval A, B. Give the values o A and B. Desiré Taylor Math 4 4

30 Sec.6 Liits Involving Ininity A. Ininity vs. DNE Recall ro section. that DNE since the unction value kept increasing. Now we will be ore 0 descriptive; any value that keeps increasing is said to approach ininity ( ), and any value that keeps decreasing is said to approach negative ininity (- ). Eaples:.) Evaluate 0 using the graph and table ethod. 0 y 0 y ) Evaluate 0 using the graph and table ethod. 0 y 0 y Desiré Taylor Math 4

31 B. A Quick Review o Asyptotes An asyptote is an iaginary line that the graph o a unction approaches as the unction approaches a restricted nuber in the doain or as it approaches ininity. Locating Vertical Asyptotes I p( ) ( ) is a rational unction, p() and q () have no coon actors and n is a zero o () q( ) n is a vertical asyptote o the graph o (). then the line q, Locating Horizontal Asyptotes. Let ( ) p( ) q( ) an b n a b n n... a... b a 0 b 0 i. I n <, then y 0 is the horizontal asyptote ii. an I n =, then the line y b iii. Eaples:.) is the horizontal asyptote I n >, there is NO horizontal asyptote. (But there will be a slant/oblique asyptote.) For the ollowing rational unctions, ind the vertical and horizontal asyptotes i any: 6 ) 4.) ( 8 g ( ) 64.) 7 ( ) 5 h 4.) k( ) Desiré Taylor Math 4

32 C. Vertical Asyptotes Vertical asyptotes occur when or The asyptote will be the line a a. a Eaple: Evaluate the it, ind the asyptote and graph the unction.) 6.).) D. Liits as Ininity A it as the doain approaches ininity: Finding Liits as Ininity o Rational Functions i. Deterine the degree o the denoinator. (Let's say degree = P) ii. Multiply both the nuerator and denoinator by P iii. Desiré Taylor Math 4 Distribute/clean up algebra and continue evaluating the it..

33 Eaple: Evaluate the it ) 4 6.).) Conclusion: For positive integers M and N such that M N. Degree o the Nuerator = Degree o the Denoinator Polynoail o Polynoail o Degree M Degree M Ratio o. Degree o the Nuerator > Degree o the Denoinator Polynoail o Degree M Polynoail o Degree N. Degree o the Nuerator < Degree o the Denoinator Polynoail o Degree N 0 Polynoail o Degree M Leading Coeicient s Desiré Taylor Math 4 4

34 More Eaple: Evaluate the it..) ) 9.) Find the horizontal asyptotes or the curve y ) Find the vertical asyptotes or the curve 4 y Desiré Taylor Math 4 5

35 5.) 6.) 7.) Desiré Taylor Math 4 6