Concept Category 5 Limits Limits: graphically & algebraically Rate of Change
Skills Factoring and Rational Epressions (Alg, CC1) Behavior of a graph (Alg, CC1) Sketch a graph:,,, Log, (Alg, CC1 & ) 1
Factoring Integer : 70 7(5)() Monomial 3 : 30 y z 5(3)() yyz Binomial : 3 1 3 3(4) 3 (two monomials) ( 3 Binomial : 3 1 3 3(4) 3 Trinomial : 6n 16n 10n 4) common factor 3 ( 4) This is Quadratic 3 ( )( ) (3) nnn (8) nn (5) n Common Factor first Always n(3n 8n 5) Quadratic n(3n 5)( n 1) Integer is a type of monomial
3 5 0 3 5 0 5 Special Factoring 5 ( 4) 5 0 5 ( 4) 5 ( )( ) 4 As long as your leading degree is even, 5 ( 4) you have try and factor 5 ( )( ) See how the powers are evenly distributed? 5 11 3 15 4 ( 11 15) If the leading degree is twice the second degree, you have to try and factor ( 5)( 3)
So how about this one? 3 8 11 4 14 Is there any common factor? No, so nothing to take out Is the leading degree twice as big as the second degree? Yes, then we have to try to factor (3 4 )( 4 ) 1, 14 or, 7? (3 4 14)( 4 1)
Special Factoring: Factoring by Grouping Remember how distribution works? ( )( 3 5) 5 3 5 10 Now, we will reverse the process 5 3 5 10 Pair-up! 5 5 3 10 Take out common factor for each pair; make sure what s in the ( ) are the same Re-group 3 5 3 5 3 5
TRY AGAIN 3 3 3 6 3 6 3 3 ( ) 3 Distribute to check if you are correct ( )( 3) 3 3 6
Think about these Quadratics Alg1] 6 7 3 6 3 Alg ] 9 5 (3)(31) 3 3 ( 1)( 5 ) CC1] 5e 3e (5e ) ( e 1) 8 4 CC1] 5e 3e 1] 3( CC ) 4( ) 15 (5 ) 4 4 e ( e 1) (3( ) 5) (( ) 3) CC] sin 3sin (sin 1) ( sin ) You need to know how to factor, but you can also use Quadratic Theorem
These are also Quadratic 11 15 ( 5)( 3) 0.5 0.5 ( 5)( 3) 7 3 0 Method : 1 1 3 7 1 1 (3 1)( ) 7 3 0 (1)( 3) 0
Special Factoring: Difference of Cubes E) 3 7 Alg method : find a root ( 3) 3 y 0 change to factor division 3 3 7 If it s a binomial, both elements are cubes: you can use this formula 333 ( 3)( 3 ) ( 3)( 3 9) The signs do not change
Special Factoring: Sum of Cubes 3 E) 16 Alg method : find a root y 0 change to factor ( ) division Common factor first ( 3 8) If it s a binomial, both elements are cubes: you can use this formula ( )( ) ( )( 4) 3 16 Again, the signs do not change
Rational Epressions: Simplification integer : 105 7 5 3 73 5 Remember: Factoring is really multiplication backward monomial : 1 y z 6 6y 3 4 1 3 yyyyzz z 3yyyyyy y If it s not monomials, you have to factor first: what are the techniques for factoring? binomial 3 3 : 6 6 3( 1) 6( 1) 3( 1)( 1) 1 3 ( 1) mied 3 8 : 3 ( )( 4) (1)( ) 4 1
More Practice: Simplify 3 1 3 ( 1)( 1) 1 (3 )( 1) 3 Know your skills Common Factor Regular Factoring Cubes Factor by grouping 4 43 4 19 ( 3)( 1) 1 ( 3)( 3) 3 1 3 3 6 6 3 3 3(4 1) 3 3( 1) 3(1)( 1) 3(1)( 1) 1 1 or 1 ( 1)( 1)
Welcome back! Do these now 1 3 3 6 6 3 3 3(4 1) 3 3( 1) 3( 1)( 1) 1 1 3( 1)( 1) 1 ( 1)( 1) 4 43 4 19 ( 3)( 1) 1 ( 3)( 3) 3
Etraneous Values What a Denominator cannot be. There are numbers that will not work in the Rational epression. 3y 1y 3 y 9y 7y 63 3y y 7 3y 7 9 y y y 9 They are -7, 3 and -3 in this case. Why?
The denominator cannot be 0, therefore y cannot be. 3y y 7 y 7 y 3 y 3 Can you see why -7, 3 and -3 do not work as answers?
Ecluded (Etraneous) Values When factor out the denominator. 3y y 7 y 7 y 3 y 3-7, 3 and -3 do not work as answers because they make the denominator zero. They are called Ecluded (Etraneous) Values
Practice: ecluded values What are the values of the variable that make the denominator zero and the epression undefined? 3 3 67 3 5 1 6 1 5 1 0 6 0 1 0 5 1 6 3 4 0 1 5 or 0. 3 30 40 3 4
Properties of Rational Functions Domain of a Rational Function { 4} or (-, -4) (-4, )
Properties of Rational Functions Domain of a Rational Function { 3, 3} or (-, -3) (-3, 3) (3, )
Properties of Rational Functions Domain of a Rational Function { 3, 5} or (-, -3) (-3, 5) (5, )
Properties of Rational Functions Vertical Asymptote A vertical asymptote eists for any value of that makes the denominator zero AND is not a value that makes the numerator zero. Eample A vertical asymptotes eists at = -5.
a h k The Parent Function can be transformed by using f ( ) a k h What do a, h and k represent? Parent vertical stretch/compression/reflection the vertical translation; VA the horizontal translation; HA : f ( ) 1 ( 1, 1) (0, undef) (1,1)
Where is an asymptote? The line where a graph approaches There can be vertical asymptotes horizontal asymptotes or What is the vertical asymptote? What is the horizontal asymptote? y 0 slant asymptotes. 1 0
E1) Compare the graphs 1 g ( ) 3 What is the transformation? Down 3 units What is the vertical asymptote? 0 What is the horizontal asymptote? y 3
Eample 1 1 g ( ) 3 What is the Domain? 0,0 0, What is the Range? y 3, 3 3,
Eample g ( ) 1 What is the transformation? Left units What is the vertical asymptote? What is the horizontal asymptote? y 0
Properties of Rational Functions Vertical Asymptote and Hole Eample A vertical asymptote does not eist at = 3 as it is a value that also makes the numerator zero. A hole eists in the graph at = 3. HA? a?
a)tell me everything you know about the functions
CC5 Learning Target Rational Functions Create a rational function equation for each : Can you identify all the asymptotes?
Can you match the functions with the graphs? What do you notice about each graph?
CC5 Learning Target Rational Functions Create a rational function equation for each : Can you identify all the asymptotes?
Can you match the functions with the graphs? What do you notice about each graph?
To Graph Rational Functions 1. Factor Completely. Vertical Asymptotes: set the denominator equal to zero and solve for the etraneous values The vertical asymptotes are the domain eclusions. 3. Horizontal Asymptotes: compare the degrees of the numerator and denominator.
Horizontal Asymptote If f( ) low deg ree high deg ree f( ) 3 5 then HA: y = 0 kind of If f( ) same same deg ree deg ree f( ) 6 3 5 look at y leading coefficient leading coefficient then HA: y = If f( ) high deg ree low deg ree f( ) 3 3 5 then there is NO horizontal asymptote Slant A.
What do you see here?
How do you graph this one?
You will need to break it down first: Long Division v.s. Synthetic Division
HA? VA? Which Quadrants? 1 f( ) 3
CC1 Review How do you sketch this one? f( ) 3 1 Rewrite HA VA Maybe one coordinate? Or - intercept? Or y-intercept?
f( ) 3 1 Solution:
Reminder: Definition of a Rational Epression In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions.
Find the Domain, Vertical & Horizontal Asymptotes, Hole, then sketch: 5 6 1 5 4 15 8 4 What re the ecluded values? These two are for practice ( )( 3) ( 5)( 3) 5,3 D : (, 5) U( 5,3) U(3, ) VA : 5 Hole : 3 HA :1 a 7 How do we get these key features?
How do you find Vertical & Horizontal(or slant) Asymptotes? Hole? How do you sketch? 56 3 3 6 53 ( )( 3) ( 3) 3 Hole at 3 a) why? b) what do you think the final graph is? ( 3) ( )( 3),3 VA :,3 HA : y 0, kind of a : cannot use so use some pts (3)( 1) VA : HA : none SA : a : use
Try Again: a] 43 VA? Hole? HA( SA)? a? b] 3 VA? Hole? HA( SA)? a?
Other than domain, range, maimum/minimum, asymptotes and end-behavior there are other key features you can consider
Error Analysis & Model Cards Error Analysis DOK Problems: If you answered everything correctly, 1 out of 3 1 out of 3 3 out of 3 + to 3- DOK 3 Problem: 3 to 4+ Turn in your Quick Check Error Analysis at the end of the class today Model Cards: Due April 19 th Thursday Pick two of the following problems based on the result of your Quick Check 3 7 1. Sketch f () VA? HA( SA)? Hole? 4. Sketch g( ) 1 VA? HA( SA)? Hole? 3. Write a rational function f ( ) that satisfies the criteria : As, y, y As 1.5 from the left, y As 1.5 from the right, y coordinates : f (1) undefined intercept at.5 Show your work and eplain your thinking
Graph Solutions to Model Card Problems 1) ) 3)
CC5: Limits
WHY LIMITS??? Conceptually, a limit captures the spirit of scientific measurement. In physical sciences, we had conceded long ago that perfect measurements are not always possible. Every ruler has some defect. But given any ruler, we can always make a more accurate one. Equality in mathematics is an etremely rigid condition. And because of what we said just above, we can't possibly hope to make good use of equality in physical contets. Limits give us the net best thing. We might set out, not to measure something eactly, but to measure it to a certain degree of accuracy. We give ourselves a certain error tolerance
Limits: Definition As the variable approaches a certain value c ( is very close to, not necessarily at, a certain number c), the variable y approaches a certain value L. Graphical Approach: Find the requested limits from the graph of the given function.
3 Ways to find Limits Graphically - eam the behavior of graph close to the c Numerically - construct a table of values and move arbitrarily close to c Analytically (algebraically)
Notations a means approaches a from the right a means approaches a from the left Constant number
Think about behavior of a graph called f() 1 y 1 y 0 y 0 y 3 3 1 y 1 y y 1 y 1 5 y 5 y 0.5 0.5
Now epress them in limits notation: ( ) ( ) Lim f 0 ( ) Lim f 0 Lim 1 Lim 1 3 f ( ) f ( ) Lim f ( ) 3 1 Lim f 1 ( ) Lim f 1 Lim f ( ) 1 ( ) Lim f 5 Lim f 5 () 0.5 0.5
f ( ) y v. s. Lim f ( ) L c There is a hole in this graph. Lim f ( ) 0 1 Lim f ( ) 0 1 f (1) NS Limits Eists even though the function fails to Eist
What does this mean????
lim Does limit eist? 1 lim 1 lim 1 f( ) f( ) f( ) 3 DNE Does Not Eist
With your partners: lim lim lim lim f ( ) f ( ) f ( ) f ( ) 3 3 1 DNE 3 0 1
Try This One Now t 1. lim 4. lim 3 3. lim 0 4. lim 6 5. lim 6. lim 5 t t t t t t 3 D 3 D
Finding limits Graphically; Numerically (by charts); Algebraically 3 g( ) Lim g( ) Lim g( ) Lim g( ) Lim g( ) 7 3 3 3 3 3 4 7 DNE 7 Lim g ( ) 3 3 Lim g ( ) 3 3 7 BTW: Do you remember how to sketch piecewise functions? ( ) Lim g 3 ( 3) 7 4 Which equation do you use based on the domain? ( ) Lim g 3 (3) 7
Now you try these: Find limits graphically and algebraically 1 1 f ( ) lim f ( ) lim f ( ) f (1) 6 8 1 1 1 4 1 g( ) lim g( ) lim g( ) f ( 1) f (1) 1 1 1 7 3 h( ) lim h( ) lim h( ) lim h( ) h( 3) 4 3 3 3 0 ( 1) v( ) 3 lim v( ) lim v( ) v() v(0) 4 1
f() g() lim f ( ) 1 lim f( ) 0 1 f (1) 3 DNE lim g( ) 1 lim g( ) 1 f ( 1) f (1) DNE
h() v() lim h( ) 3 lim h( ) 1 lim h( ) 3 0 h( 3) 0 DNE lim v ( ) 1 lim v ( ) 4 1 v() 1 v(0) 1
Happy Tuesday!!!! 1) t t Lim t 4 ) Lim 5 5 5 3) Lim 3 101 9 4) Lim 3 101 9
Finding limits algebraically 1) t t Lim t 4 ( t)( t) t t 4 ) Lim 5 5 5 5 1 ( 5)( 5) 5 5 1 1 5 10 3) 4) Lim 3 Lim 3 101 101 9 9 ( )( 3) ( ) ( 3)( 3) 3 ( ) 3 ( 3 ) 1 3 3 3 X=3 is the VA, so limits as approaches 3 DNE
How do you find limits for these functions? 1 4 4 1 1) lim 1 ) lim 4 1 3 3) lim 4 9 7 4) lim Never assume limits do not eist
You can solve the limits by graphing 1 1 1 1) lim ) lim 1 1 4 4 4 1 3 1 9 7 1 3) lim 4) lim 4 4 3 7
But you should also be able to solve algebraically: 1) lim 1 1 1 1 1 1 ( 1)( 1) You can rewrite any radical rational epression using rationalization Remember conjugation? 1 ( 1)( 1) 1 1 lim 1 1 1 1 substitution : 1 1 1 1
3) lim 4 1 3 1 3 4 1 3 What s the conjugate? 1 3 1 3 1 3 ( 4)( 1 3) 1 4 1 3 1 3 13 ( 4)( 1 3) 4 1 ( 4)( 1 3) 1 3
Finding limits algebraically What do you remember about comple fractions?
1 1 ( ) 4 How do you add, subtract fractions? Common Denominator Lim 1 4( ) 1 4( ) 1 16 1 4 1 ( ) 4 4 1 4 1( ) 1 * 4( ) 1 1 * 4( ) 4( )
1 1 Lim 1 ( ) 1 1 1 Lim 1 1 1 1 Lim 1 1 Lim 1
Lim 3 7 4 3 0 0 8 0 is not no solution( DNE) it ' s a real number! Lim 3 1 9 1 4 Lim 0 1 1 3 3 9 Lim 0 1
Solve algebraically (you have average 4 minutes per problem): 1) Lim(3 5) ) 3) 4) Lim 3 Lim 9 Lim 3 7 3 3 9 1 1 We will review the concept f continuity after the practice On Monday: we will have a mock eam, then Mastery Check on Thursday
1) Lim(3 5) 3() 5 7 3 7 ( 3)( 3 9) ) Lim 3 9 (3) 3(3) 9 7 3) 3 3 3 3 3 3 9 1 1 1 9 9 3 ( 9)( 3) 3 9 3 6 Lim 9 4) Lim () 4 1 1 1 1
Continuity
Definition: Continuity A function without breaks or jumps The graph can be drawn without lifting the pencil 89
A function can be discontinuous at a point A hole in the function and the function not defined at that point A hole in the function, but the function is defined at that point
A function can be discontinuous at a point The function jumps to a different value at a point The function goes to infinity at one or both sides of the point, known as a pole
Definition of Continuity at a Point A function is continuous at a point = c if the following three conditions are met 1. f(c) is defined. 3. lim f( ) eists c lim f ( ) f ( c) c = c
These functions are dicontinuous But these two have limits at -> c
"Removing" the Discontinuity A discontinuity at c is called removable if If the function can be made continuous by defining the function at = c or redefining the function at = c 94
Do These Now (1 minutes) Based on your Mock Eam s result: what do you need to practice on? 9 1) Given f ( ) a] Sketch b] Find Lim f ( ) algebraically 3 3 Remember the DOK 3 from yesterday ' s Mock Eam? ) Lim 4 4 1 1 4 3) Lim 7 3 7
Solution: what do you need to add to your notes? 1. a] b] Lim 3 9 3 ( 3)( 3) 3 3 ( 3) 3 6 Lim f 3 ( ) 6 4 ) 1 4 1 4 4 4 4 4 4 4 4 4 Lim 4 4 4( 4) 16 3) 3 3 7 3 9 ( 7)( 3) Lim 7 1 3 1 1 1 3 7 3 6
Where are you now and what do you need to work on? Group/Pair Practice: Yellow: find limits graphically (DOK1) Green & Pink: find limits graphically and algebraically (DOK). Answer Key at the back of the papers * Which level do you think you are at right now? Print your name and check one of the levels.
DOK1 + + ANSWER a) 1 b) c) DNE d ) e)0 f ) DNE g)1 h)3
DOK : Limits of rational functions Find the limits graphically and algebraically: 1) Lim 3 73 3 8 8 8 1 7 3 7 3 ) a] Lim b] Lim c] Lim 3) a] Lim b] Lim 0.5 1 3 1 3 3 4) a] Lim b] Lim 4
( 1)( 3) ( )( ) 1) y ) y ( 3) ( 1)( ) 7 3 8 8 Lim 3 3 8 8 b Lim DNE c Lim 1 1) Lim 5 ) a] 3 ] ]
( 1)( 3) 3 3) y 4) y 1 ( )( 1) 7 3 3 0.5 1 7 3 3 b Lim 1 3) a] Lim DNE 4) a] Lim DNE b] Lim 8.4 ] 0.7 3 4
DOK : Limits of special rational functions Limits of piecewise functions Find the limits graphically and algebraically: 1 1 5) a] Lim b] Lim 0 1 1 1 1 1 6) a] Lim b] Lim 0 1 1 1 7) f ( ) Lim f ( ) f () 6 5 8) g( ) Lim f ( ) f (5) 10 5 5
5) 6) 1 1 5) a] Lim 1 6) a] Lim 1 0 0 1 1 1 1 b] Lim 0 b] Lim 1 1 1 1
6) 7)
DOK: Finding limits algebraically if limits eist Do #19~0, 6~30 first
Answer key for odd numbers, even numbers answers will be provided later 19)5 1) DNE 6 3) 5)4 5 1 1 7) 9) 6 16 1 31) 9