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**MHF4U - Grade 12 Functions - Homework - Fall, 2013**
Review of functions from Grade 11 math class - general forms, sketch, significant points Find significant points and sketch each of the following: y = x² + 6x + 8 y = 1 / (x-2) y = √ x + 4 - 1 || Textbook distribution Syllabus and Timeline
 * =**Today's Date**= || =**Class Topics and Notes Summary**= || =**Homework/Handouts**= ||
 * Sept 3 || How to succeed in this course - course syllabus and timeline (see main page), grading policies (most consistent, most recent, lates/zeros), how to earn your mark (7 tests worth 55%, [|assignments worth 15%], final exam worth 30%), the importance and value of homework and quizzes. Attendance policy.

FunctionReview

Do two of the three problems Show some work in finding significant points

Share a google doc with your teacher - give edit permissions || y = a f(x-p) + q. So, if you can sketch y = x², you should be able to sketch (and find significant points for) y = -3x² + 18x - 15 Similarly, if you can sketch y = √ x, you should be able to sketch y = 3√ x + 4 - 9 What about y = 1/x and y = -1/(x - 3) + 2 ? Homework problems: 1. sketch each function below using transformations 2. then find intercepts, vertex/origin and asymptotes using algebra || Test next Wed on Gr 11review
 * Sept 4 || Review of yesterday's homework - required skills of factoring, completing the square, sketching general functions and their transformations. If we know the general shape of y = f(x), we should be able to locate and sketch

a. y = -2x² - 8x + 10

b. y = 1/(x+4) - 3

c. y = -2 √ x - 1 + 6 || Find the inverse of y = (x + 3)² - 2 and of f(x) = 1/(x+4) - 3 Domain and Range - how to write for this class - x € R, x ≥ -2, y < 12, y ≠ 3 Review of everything so far. Can you find all significant points and sketch the functions below? Can you find the inverse of each function below? What are the domain and range of each function? What does the transformed graph look like? Equations of Asymptotes - x = #, y = # Best format for domain/range: {x € R | x > -4} {y € R | y ≠3} || y = -2 (x + 3)(x - 1) y = 1 / (x-3) + 2 y = - √ x + 3 - 4 || Quiz - everything we have done so far - base functions, transformations, intercepts and other important points, asymptotes, domain and range, inverse of a function || Identifying Equations MHFquiz1 || Review of everything taught/learned/relearned so far 1. **sketching** a. by finding intercepts, asymptotes, origin b. with transformations (graph based on left/right/up/down shift, inverted vs right-side-up, A-value = pos/neg) 2. **domain/range** a. determine the domain and range of a function, using proper notation {x€ R | x< 0}, {y € R | -4 < y < 10} b. determine the domain and range of a function based on transformations 3. **inverse of a function** a. sketch a function and its inverse (based on transformations and the line y=x) b. calculate the inverse of a function (switch x and y, solve for y) Do this for three functions - root, quadratic, and rational Intro to the Absolute Value function - what is it, what is the notation, how does it affect functions? Compare the graphs of y = 2x + 1, y = |2x + 1|, and y = |2x| + 1 Compare the graphs of y = (x + 2)² - 3, y = |(x + 2)² - 3|, and y = |(x + 2)²| - 3 Compare the graphs of y = √ x - 2 - 3, y = |√ x - 2 - 3|, and y = |√ x - 2 | - 3 || On your google doc paste ONE fooplot image with the functions below. Resize the window so that the significant points on each function are visible. Make sure the function names are visible Then find the significant points for one of them, the domain and range for another and the inverse for the last. The work does not go on the google doc. it goes in your homework
 * Sept 5 || Review of last year - finding the inverse of a function. Switch x and y, solve for y, call the result f -¹ (f-inverse)
 * Sept 6 || Identifying equations based on graphs - handout - you will see this on the test next week
 * Sept 9 || Collect quiz - to return tomorrow
 * end of test material for Wednesday **

f(x) = 2(x+4)² - 8 f(x) = 3/(x - 4) - 3 f(x) = -2 √ x + 3 + 6 || Sketch the following absolute value functions y = |x + 2| y = |-2x + 4| y = |(x-2)² - 5| y = - |(x-2)²| - 5 y = |1/(x+3)| y = - |√ x + 3 - 4| ||  || Graphing problems on board: 1. sketch f(x) = x² - 4x - 5, f(x) = |x² - 4x - 5|, and f(x) = |x² - 4x| - 5 what is the difference between each of the graphs? how does the abs value affect each graph? 2. do the same for f(x) = 1 /(x-2) - 3, f(x) = |1 /(x-2) - 3|, and f(x) = |1 /(x-2)| - 3 what is the difference between each of the graphs? how does the abs value affect each graph? 3. do the same for f(x) = √ x - 2 - 3, f(x) = |√ x - 2 - 3|, and f(x) = |√ x - 2 | - 3 4. generalize your results, based on what you have graphed: what's the difference between y = f(x) - b, y = |f(x) - b|, and y = |f(x)| - b 5. predict AND demonstrate what happens in each case for y = f(x) + b  If you need additional practice, try the handout, but I don't want to give it to you since it uses tables of values instead of transformations! || worksheet
 * Sept 10 || Absolute Value
 * Sept 11 || Test - whole period ||  ||
 * Sept 12 || Review of absolute value functions. We will learn later that these are examples of COMPOSITE functions: a function within another function.

p 16 # 1 - 10 || Sketch y = -¼ x² + 4 and calculate the slope of the line between x = 1 and x = 3 (we call this line a secant) You will estimate the slope of this secant by taking values of x closer and closer to 3 so we calculate the slope between x=2 and x=3 (finding the y-values first) then we find the slope between x=2.5 and x=3 we keep doing this until the two points are 0.01 apart (x=2.99 and x=3) at this point we have a really good estimate for the slope of the tangent at x=3 (m=-6) *when you take calculus, you will find a much easier way to calculate the slope of a tangent to a function Rates of change - work period - finish three sections of graphing for homework || ratesOfChange
 * Sept 13 || ** Introduction to Rates of Change ** - how quickly is the slope of a secant or tangent changing?

answers: m = 6 m = -1/9 m = 1/2√3 || height of object given by h(t) = -5(t - 4)² + 80, t=sec, h=metres value of automobile given by v(t) = 20,000 (.9)media type="custom" key="23832394" t=yrs, v = dollars In each case, find the average rate of change between t=1 and t=3 AND the instantaneous rate of change at t=3 (which requires a second point 0.01 away) Review of Rates of Change - finish worksheet from last class then work on p87 word problems. Remember that we are always looking for one of two things: - the average rate of change (between two points some distance apart) - the instantaneous rate of change (at some exact point) for the instantaneous rate of change we take a second x-value that is 0.01 units away || P87homework || What if x gets very close to a vertical asymptote (undefined region)? And we will learn new terminology and notation to deal with limits. [] || p 25 # 12 - for each function: sketch domain/range end behaviour || Short discussion of: symmetry, increasing/decreasing regions, continuous and discontinuous - from the first day handout || properties of functions
 * Sept 16 || Solve two problems on board with average and instantaneous rates of change
 * Sept 17 || ** End behaviour of functions ** - introduction to limits. We are asking "what happens to f(x) as x becomes very large? what if x becomes very small?
 * Sept 18 || Review of End behaviour

chapter review problems || relation to speeding fines (hwy 6, south of 401) - what is the relation between your speed and the amount of the fine? defn: a function defined by two or more conditions along the domain of the function Take up introductory sheet - make sure you can define the function and the domain for each piece of the function Second handout of piecewise functions - do the first half for homework create a graph for each problem and define the function with algebra || Piecewise1
 * Sept 19 || ** Introduction to Piecewise Functions **
 * Today we learned how to DEFINE a piecewise function - with domain. **

do problem 2 from the piecewise handout

then do two more columns from the chapter review problems || ex: phone company charges $0.25/min for first 10 min, $0.10/min for next 10 min, $.02/min for anything beyond 20 min sketch a graph of this piecewise function, find an equation for each part (remember to subtract some amount of minutes each time) ex: the electricity rates in a city are dependent on both the time of day and the amount of elecricity used. Residential customers are charged $35.00 per month PLUS $0.08 per unit (kw hr) for the first 600 units; they are charged $0.10 for the next 400 units, and $0.12 for anything over that amount. sketch a graph, locate and label the endpoints of each piece, and determine the piecewise function - including the domain ex: courier company - charges $10.00 per package, plus $0.20/kg for first 5 kg, plus $0.15/kg for next 5 kg, plus $0.05/kg for everything over 10 kg (how is the graph the same? different? what are the equations? what are the domain restrictions for each part?
 * Sept 20 || Review of piecewise functions - new examples

Work period for second piecewise function sheet, the last columns of the chapter review, and the quiz. Quiz is due Monday, if you want the feedback and the points. || Piecewise 2

Quiz 2

Test next Wednesday!! || Review of piecewise function handout - both applications and math functions Chapter review handout - can you find the necessary information? Review of key ideas for Wednesday's test How to use fooplot to graph a piecewise function - you need this skill for the summative assignment for piecewise function - example 3 - the code for the function follows this format: (condition1)*(piece1) + (condition2) + (piece2) + ... (x<=-2)*(3)+(x>-2&&x<2)*(x^2-1)+(x>2)*(2*x) every condition has an inequality or a boolean operator (&& means both conditions must be met) Introduction and handout of Summative Assignment # 1 - due next Monday || Finish work on
 * Sept 23 || Collect Quiz - make sure I get it by the end of the day. You get it back tomorrow.

piecewise functions - 2 sheets function review - chart

[|Summative # 1] || Look at the relationship between the highest power of a polynomial, the general shape of the graph, and the end behaviour of the function. Note: finite differences allow us to determine the degree (highest power) of a function through a series of subtractions; when the differences are constant, we have found the degree of the function. Start the assignment sheet on finite differences and the problems from text on the sheet
 * Sept 24 ||  ||   ||
 * Sept 25 || Test on Unit 2 - whole class ||  ||
 * Sept 26 || ** Introduction to ** ** new unit on polynomial functions ** - unit outline and review of finite differences.

Polynomials - degree, max number of turning points, end behaviour Graphs of polynomials in factored form Determining the polynomial from the graph graph using x-intercepts and end behaviour (use +/- method). The degree of a polynomial will tell you about the end behaviour - it will be useful to learn this pattern. We will deal with polynomials in BOTH standard form and in factored form.

explanation of single, double and triple roots: - single root = regular x-intercept - double root = turning point on x-axis - triple root = inflection point on x-axis = graph changes from concave upward to concave downward

Based on what we already know: - the combination of "A" and the polynomial degree tell us the end behaviour of the function - the factored form gives us the x-intercepts, roots, or zeros - standard form gives us the y-intercept - the degree of each factor (single, double or triple) tells us the intercept behaviour at that point || ch3 Outline

p 15,16 # 9, 14, 15

Differences

graphs

equations || finish work on graphing polynomials with single, double and triple roots - make sure the inflection point is easy to distinguish from an intercept (single root) ||  || ex: f(x) = x⁴ - 4x³ + 3x² + 4x - 4 divided by x-2 ||  || how to determine the first factor? we can guess, but we can also look at the last number of the polynomial - if the last number is 6, then (x-1) (x-2) (x-3) are all possible factors (and there are more). Josh Mak found a relationship between the numbers in the factors and the coefficient of x2 term ||  || New: is there a way to factor a polynomial without doing Long Division - which may or may not result in a zero remainder? We will learn the FACTOR THEOREM and only use long division when the factor theorem gives a remainder of zero. Factor Theorem: (x-a) is a factor of f(x) if and only if f(a) = 0 Remainder Theorem: if f(x) is divided by (x-a), the remainder is f(a) The remainder theorem is not very useful because we are only interested in dividing when the remainder is zero. So, when f(3) gives a non-zero remainder, we know that x-3 is not a factor; however, it doesn't tell us what IS a factor. For y = x³ - 5x² + 3x + 9 - show that x+1 is a factor by evaluating f(-1), which equals zero, right? Then perform synthetic division to reduce the cubic function to a quadratic, which is easy to solve. In the end, you get y = (x-3)²(x+1) - which is easy to sketch, and find the end behaviour, domain and range. || p 177 # 6 factor two cubics and one quartic || Intro to polynomial inequalities - when is a function above or below the x-axis? (or, in harder cases, above or below another function)? The key points for this lesson are as follows: - we are trying to determine when one function is above or below another function (including the x-axis) - you can graph the two functions and then determine where the functions intersect, but that's a lot of work - it is (in general) easier to solve for the points of intersection algebraically and then use test points to determine which y-values are greater (or less) in that interval. - we organize the test points in a + / - chart; we don't actually care what the y-values are, we just care which one is above or below The first two pages of the handout are the assignment; the last two pages are graphs that show the intersection. The graphs are meant to verify your algebraic work, not to replace it. Textbook reference for additional practice - ex 4.3 p 219-228 || Factoring Strategy Polynomial Inequalities
 * Sept 27 || Work period for polynomials and for first summative assignment- which is due Monday
 * Sept 30 || Christian College fair - one class started with long division of polynomials by binomials
 * Oct 1 || Continue with division of polynomials
 * Oct 2 || Review of finite differences, factored form of polynomials, roots, end behaviour and graphing.
 * Oct 3 || Review of our factoring strategy for polynomials - cubics and quartics

[|Here is the link] to the presentation try the new online graphing calculator at Desmos.com

B3 and C1 or C2 - sketch - solve for intersection points - use +- chart to find regions where f(x) >,<, >=, or <= g(x) || Today we will review the concepts of polynomial inequalities - the main question is: when is one function above or below (greater than or less than) another function? We went through [|two examples in class today and they are here] I've included a textbook explanation for some of you who may have missed my class 3 or more times this week - make sure you keep up with homework. ||  || Last topic - families of functions - can you sketch polynomials that have the same characteristics? ie: roots? Quiz handed out today - due tomorrow Test is this week - Thursday or Friday - you choose, but you must write it by the end of the week. No exceptions, please. || polynomial quiz end behaviour families of polynomials || Introduction to polynomial regression - determine the function from given data. We use [|Xuru's website] for this The question we are solving is this: given a set of data, can we find the best polynomial function that models the data? We might be unsure whether to use a cubic or a quartic function, but there is an error measurement - called the RSS value - the residual sum of squares - which tells us how far each of the points is from the function we picked. The lower the RSS value, the better the data fits the function. An RSS value of 0 is a perfect fit. || polynomial Regression
 * Oct 4 || Many of you are gone today and for the weekend. I'm probably moving the unit test to Thursday or Friday - depending on what else is scheduled.
 * Oct 7 || Review of last week's work - polynomial inequalities
 * Oct 8 || Review for test - if you need more practice

test review problems p 184 # 3,6,10a,11d,15b,c

inequality problem: when is f(x) = x³-4x²+3x+10 less than or equal to g(x) = x²+4x+5 use a +/- chart or a sketch of the new function in your answer || numerator and denominator, but we will move quickly to functions that are quadratic in both numerator and denominator. For the function y = f(x) / g(x) - solve f(x) = 0 to find x-intercept - solve g(x) = 0 to find vertical asymptotes - let x = 0 to find y-intercept Start with reciprocal functions ie: f(x) = x+2 and g(x) = 1 / (x+2) what are the similarities and differences? what does the the graph of f(x) tell me about the graph of g(x)? can I use the y-values from f(x) to find the y-values for g(x)? - what about the functions f(x) = x² + 3x - 4 and g(x) = 1/f(x)? Can I show that y = 1/(x+2) - 3 and y = (-3x-5)/(x+2) are the same? The first one is much easier to sketch, so we want to always default to transformation format: y = 1/(x-a) +b || p. 255 # 2a,b,c,3, 6 for # 6 do not verify by using a graphing program. Find the intercepts and make a sketch of the reciprocal function
 * Oct 9 || New Unit - Rational Functions - any function that contains a variable in the denominator. We will start with rationals that are linear in both

p 262 # 2c,g - write in transformation form, then sketch || If you need more time, you should come to class 10 min before the end of lunch. Don't ask for extra time once the final bell rings. ||  || Keep in mind: - your data is different from that of your classmates - your goal is to convince me (or someone who missed the unit) that you can do and explain the work - if xuru gives you a number like 3.2e-12, the number is so small that we can ignore it - if the RSS value is large, we need to try a different degree; if RSS=0, the fit is exact. || [|summative assignment] - polynomials || 1. sketch y = (-2x - 5)/(x + 3) - if this were in "standard" rational form, it would be easy to graph, right? like y = 1/(x+3) - 2 2. sketch f(x) = (2x-6)/[(x+4)(x-1)] - find y-int by setting x = 0 - find x-int(s) by solving f(x) = 0 (we use 0/1, since 0/0 is undefined, right? - it's called indeterminate form) - find vertical asymptotes by solving the denominator = 0 (related to the x-ints of the reciprocal) - find the limits on BOTH sides of each asymptote (use the +/- method we have used in previous lessons) - verify with a graphing program (fooplot/desmos) || write the following functions as transformations, then sketch and verify the intercepts and end behaviour with calculations: y = (x-2)/[(x-1)(x+3)] y = (3x+5)/(x+2) y = (2x-6)/(x-4) y = (1-x)/(x-3) y = (4x+5)/(x+1) || 3. sketch f(x) = (2x-6)/[(x-4)(x+1)] you might think the graph has just moved, but it isn't quite that simple 4. sketch f(x) = (x+1) / [(x-1)(x-5)] 5. sketch f(x) = (x-4) / [(x-1)(x-5)] look for similarities and differences between the functions based on the location of the intercept(s) relative to the vertical asymptotes || 1. y = (-2x+3)/(x+2)
 * Oct 10 || You are taking your test today or tomorrow - each class meets once right after lunch.
 * Oct 11 || For the other day - you are going to work in the commons OR in the library. You have a summative assignment to work on - it is based almost entirely on the skills we have learned in this unit plus the lesson from Tuesday on polynomial regression. It is due next week Friday, so you can get much out of the way by working ahead of schedule.
 * Oct 15 || Review: sketching rational functions where both numerator and denominator have x terms:
 * Oct 16 || Rational functions: Based on our work with problem 2 from last class, can we efficiently find the required points/limits so that we can sketch a more complex rational function?

2. y = (x+4)/[(x+2)(x-1)]

3. y=-(x+2)/[(x+1)(x+3)] || a. x-int = -1, y-int=pos, v asymp = -3,1 b. x-int = -2, y-int=pos, v asymp = -1,2 c. x-int = -2, y-int=neg, v asymp = -3,-1 d. x-int = 4, y-int=pos, v asymp = 1,3
 * Oct 17 || What If you are given intercepts and vertical asymptotes, can you sketch the graph and determine the function?

New: Using the relationship between the intercepts and vertical asymptotes, can we make an educated guess as to the shape of the rational function for any linear numerator and linear or quadratic denominator? || Rational Quiz

Rational Sketching

Rational Applications || - we need to know the general shape of the graph - we are (primarily) interested in the graph when x>0 (since most apps are time-dependent) - many of the questions are asking for intercepts or asymptotes, without saying so directly Quick sketch method for rational functions - using what we know about dominant terms (end behaviour), limits near vertical asymptotes, and intercepts, can we sketch the rational function without completing the +/- chart for vertical asymptotes? Work period - quiz available - due Monday ||  || If you get your quiz to me before I leave school today (or first thing tomorrow morning), I will have it for you by class time on Tuesday. We will review work from the past few days during this class. If time permits, we will start rational inequalities, which is not on the test. || Your summative assignment is still due today; even if you are at the university presentation. || Inequalities with rational functions (NOT ON TEST) - when is f(x) above/below a constant value? - when is f(x) above/below a linear (or rational) function? we will restrict our work to situations in which you don't need to solve anything more complicated than a quadratic equation General Method 1. sketch the functions - transformations, intercepts, asymptotes (quick, not too detailed) 2. solve for the point(s) of intersection 3. determine the vertical asymptote(s) of the function(s) 4. simplify the inequality - move everything to one side, make common denominator, add/subtract as required, simplify (see problem # 1) 5. use the +/- chart with the regions defined by the points of intersection and the vertical asymptote(s) Handout with 6 problems - first and second are done for you, you should do at least two on the back of the sheet - rational/linear is expected, rational/rational is good for those going to calculus. || Rational Inequalities || With the handout, you need to draw a line between two known points on f(x) and then find the equation of the line g(x). Then go through the steps (algebra) to find out when f(x) is <, >, <=, or >= g(x). The hardest part of this exercise is the algebra - specifically, making a common denominator so that you can factor f(x)-g(x) = 0. || [|Handout] || Review of grade 11 trig - primary trig ratios: sin, cos, tan; families of angles; special triangles; CAST rule; solving for two angles that have the same trig ratio; graphs of sin, cos, and tan functions || MHFTrigIntro || where A = amplitude, k=horizontal compression or expansion, h = horizontal shift, and v = vertical shift. We ALWAYS verify a non-zero point on the transformed function to see that it matches the sketch. Ex:Temperature varies between -4 at 2 am and +8 at 2 pm. Sketch one cycle of this, starting at 12 am. Then determine the function that best fits the curve AND verify a non-zero point on the graph with your function. The hardest part of this is determining the stretch/compression factor. we use the relationship k = 360/(time for one cycle), which gives k= 360/24 = 15. What does the 15 mean in this case? Think of it this way: if sine normally takes 360 units along x for one complete cycle, in this application it takes only 24 units for one cycle; that means we get 15 complete cycles in the space of 360 units? Does that work for you? Similarly, if k=3, we get three waves in a space of 360 units, which means we get a complete cycle in 120 units (or degrees) || [|Trig Graphs and Applications] || We use trig to find the first angle that gives -0.4 as a solution and then we use symmetry to find the second solution. Introduction to Radians - think of this as a different way of measuring angles, much like celsius and fahrenheit are different ways of measuring temperature... or kilograms and pounds are different ways of measuring mass/weight. ONE radian is the angle at which the arc length (around the circumference) is equal to the radius - see this picture here is a unit circle for you to fill in with radians (degrees are given) There are exactly 2¶ radians in a circle, so 360 degrees equals 2¶ radians. From this relationship we can determine the other angles in radian measure. We focus on the 30, 45 and 60 degree families - now called ¶/6, ¶/4, and ¶/3 families. You will need to practice converting from any number of degrees to radians - the answers will only work out to nice fractions if the original angles are in one of these families. || One more review sheet from last year's work
 * Oct 18 || Applications of Rational Functions - use graphs and features of rational functions to answer questions. Keep in mind that
 * Oct 21 || University of Waterloo is here period one. I will be available all day for those who have specific questions for Wednesday's test. Yes, it is still on Wednesday - as we agreed last week. You do NOT want to put this test off until you return from Washington.
 * Oct 22 || Last chance for questions on tomorrow's test
 * Oct 23 || Test on Rational Functions - whole class ||  ||
 * Oct 29 || Review of Rational inequalities - from last week.
 * Oct 30 || Introduction to Trigonometry
 * Oct 31 || New: transformations of trig functions - the goal is to solve application problems. We use the general form f(x) = A sin k(x-h) + v,
 * Nov 1 || Review of solving for two angles that have the same trig ratio. It is fine when the ratios are from the special triangles, but what do we do when we have to solve sin(x) = -0.4, which is not a ratio on as special triangle?

p 321 # 3,7,8 (only do the positive ones) p 330 # 5,6 (only solve the primary trig ratios - sin, cos, tan) || - convert angles into radians - solve problems like: when is sin(x) = -1/2 or what is cos(7¶/4) New - solving trig equations using radians - see handout. - start with 1st quadrant angle - ignore the negative sign (if there is one) - use special triangles and CAST rule to find the angle or angles; or to find the ratio - every question should have two answers New: linear speed and angular speed of rotating objects Two calculations - angular speed and linear speed - use example of a wheel turning, let's say 15 times per second. If the wheel has a radius of 0.3 metres, we can calculate: - angular speed = 15 * 2¶ = 30¶ rad/sec - linear speed = since 1 rad equals 0.3 m of arc length, 30¶ rad gives a distance of 9¶ metres in 1 second (or 102 km/hr). || Finish p 330 # 5,6 (only primary trig ratios) soving trig equations using radians
 * Nov 4 || students who were absent on Friday need to spend a few minutes reviewing how to:

handout || - can you determine two functions that describe the graph? one a sine and the other a cosine? Again, the hardest part is finding the k-value, which is 2 ¶ / (time for one cycle) ||  || Graphs of trig functions - determine the equation that best fits the graph. Hardest thing is to find the k value for horizontal compression/stretch. || Do the first half of the handout
 * Nov 5 || Take up first half of Ferris Wheel handout - hardest parts are to calculate the k-value and to find when the height of the rider is 12 metres or higher. This requires solving the equation for t, given y = 12.
 * Nov 6 || Reminder - Test next Wednesday - only radians. Solving trig equations using special triangles, families of angles and CAST, Angular speed and linear speed (radian applications), and applications of trig functions (tides, wheels, temperature applications)

finish at least two graphs from the new handout

If time permits, finish the ferris wheel handout || Your task is to translate the given information into those quantities. Some things to remember in solving these problems: - make sure your time units agree - if the angular speed is in rads/sec, your linear speed should be in metres/sec, rather than km/hr - a linear speed can be used to find a distance along the circumference - a speed of 90 km/hr tells us that the object travels 25 m along the circumference in 1 sec. - many of these problems can be solved by applying ratios: if the angle is 1 rad, the arc length and the radius are equal; if the angle is .05 rad, the ratio of arc:radius is 1:20 - it may be useful to use the formula arcLength = radius * Ø || - due tomorrow
 * Nov 7 || Review of radian applications. in solving radian applications we are always given at two of the following: rate at which the central angle changes (rad/sec), rate at which the distance along the circumference changes (m/sec or km/hr), and the radius of the circle.



|| Summative Assignment - due Monday as well Today is a work period for trig and radian applications Here is a good test question: the outside temperature varies between -6 and +4 degrees, with the coldest temperature at 2 am (and the warmest, 12 hrs later). Sketch one cycle of this relationship, starting at 12 midnight. Then determine both a sine and a cosine function that match the graph - and verify a known point on the graph. Lastly, use your algebra and symmetry skills to find when the temperature is less than -3 degrees. this is a good test question. the last answer is: at 6.4 hrs and at 21.6 hrs, so from about 9:35 pm to 6:25 am ||  || Review of trig applications for test. 1. 2. textbook problems - p 361 # 6,7. In addition to the given questions, answer these: - when is the temperature (#6) = 110 degrees? - when is the frequency (#7) = 600 hz? ||  || You may use the unit circle handout with radians and families of angles for this quiz. you will not be allowed to use the sheet on the final exam. Two radian formulas that will be given are: A = r Ø (lengths and radians) and v = r ω (linear speed and angular speed) ||  || 1. Introduction to Trig Identities - can show these to be true for specific values, like pi/4 or 2pi/3 - we need to show that these statements are true for EVERY value = general solution. We showed that these two statements are always true, so we can use them to prove other identities. We also use (for the first time this year) the reciprocal trig functions - cotangent, secant, and cosecant. 1. sin² Ø + cos² Ø = 1 2. tanØ = sinØ/cosØ 3. secØ = 1/cosØ 4. cscØ = 1/sinØ 5. cotØ = 1/tanØ The bottom of the handout explains a few key steps to successfully proving trig identities. The key ones are: replace tan Ø, secØ, cscØ, and cotØ, combine terms on each side of the equal sign to get one term, and NEVER move terms from one side to the other. NEVER. 2. Trig Regression using Desmos - see handout ||
 * Nov 8 || Quiz - hand in no later than Monday for Wednesday's test.
 * Nov 11 || Quiz due today; summative assignment due today.
 * Nov 12 ||  ||   ||
 * Nov 13 || Test on first half of trig unit. Make sure your neighbour's calculator is in radians, not degrees.
 * Nov 14 || Two things for this class - intro to trig identities, manual regression techniques for periodic data.

do the first four identities on the handout. We will work on the back side of this sheet next week. Do not attempt #18 or #22 before completing the rest of the questions. You've been warned.

|| 1. Trig Identities 2. Compound angle and double angle formulas 3. Trig Identities using the formulas || Trig Identities handout Ex 10.8 # 4, 5,7,8, 9,11,12,13,14 || We have already done some of this in class- temperature in Buffalo, sunrise in Washington. The goal is to determine the best trig function to model a real-life phenomenon. Addition and Subtraction formulas Add/Sub -[] Double - [] - we will only work with the sine and cosine formulas - you do not need to memorize them - you must be able to solve problems with them AND use them in trig identities ||  || Take a problem like this: evaluate sin (5 ¶/12) using the addition/subtraction formulas and the special triangles. 5π/12 can be written as π/4 + π/6 (= 3π/12 + 2π/12) Now use the addition formula to write sin (π/4 + π/6) as .... then use the special triangles and CAST to evaluate the expression
 * Nov 18 || Because several of you are leaving for OFSAA tournaments on Wednesday, I will teach much of content for this unit on Monday and Tuesday. Then Wed-Fri can be used for completing a lot of the homework, working on computer, and starting the next summative assignment.
 * Nov 19 || Half of today's class is for learning trig regression using fooplot and desmos. We will start with the last set of data from the handout.
 * Nov 20 || Today we will do work with the addition and subtraction formulas - you do not need to memorize the formulas, but you need to know how to use them.

If you'd like more explanation or proof of the addition and subtraction formulas, [|see this link] on Kahn Academy || - the handout you received in class has you determine some of the angles in the addition and subtraction formulas. ex 10.8 # 15,16,17,20,21 compound angles #1 all answers on bottom of sheet. || 10.8 # 23, 25, 27 Ex 7.4 (p.417) # 9-11 - we did 10e and 11L in class -focus on problems with sin and cos components || Here is a list of cities by latitude - [] || || - chromebooks - finding the sunrise/sunset data and temperature data for your chosen city. Make sure you get a copy of the handout. - Trig identities from the handout (ex 10.8) - Trig compound and double angles - be able to calculate the exact values (Nov 20 handout) and use the compound and double angles in trig identities Tomorrow you will have a quiz on the second half of this unit. We will also do a bit of work with solving trig equations and inequalities. ||  || Begin Trig equations - the difference between equations and identities is that identities are ALWAYS true, but equations are true only for SPECIFIC VALUES. Worked through the Matching Cards handout - this highlights the different forms for trig equation problems. Finish the first side of this handout || || Solving trig equations and inequalities - reference is 7.5 and 7.6 in the textbook Work through the four problems on the new handout - trigEquations 2 - and then start working on the back side of the Matching Cards handout - # 6-16 || || on trig identities and compound/double angle formulas quiz is due on Monday || Homework from trigEquations2 Given Sin A = 3/5 (in first quadrant) and Cos B = -12/13 (in second quadrant), can you determine the value of Cos(A+B) by: -draw the location of the angles on an x-y graph -determine the values of Cos A and Sin B from the diagram -using the expansion of Cos(A+B) and evaluating the expression || Finish Compound Angle handout - part 2. Pay attention to which quadrant the angle is located in - CAST p 401 # 9,10 p 407 # 4-7 p 427 # 6, 8, 9, 13, 14, 18 p 436 # 6-10 || Summative # 4 due date is Friday - many of you will have Thursday to finish that work. There will be two more summatives available - one on Exponential Functions and the last is the exam review. Begin last unit - Exponential and Logarithmic Functions Graphing exponential and logarithmic functions - 2 ways 1. use transformations of base exp or log functions 2. use algebra to solve for intercepts, asymptotes, and end behaviour || - outline, graphs We worked on the questions for exponential functions, not logarithmic functions. Unit test on Trig Identities and equations - whole class ||  || The same rules apply as in finding other inverse functions: switch x and y, solve for y, call it y-inverse The only new piece of the puzzle is that the inverse of an exponential function is a logarithmic function. If the base of the exponential function is 3, the base of the log function is also 3. So we can show that the inverse function of y = 2^(x+3) - 1 is y = Log(base 2) (x+1) - 3 Review the two key log rules: log a c = log c / log a if log a c = x, then a media type="custom" key="24653790" = c || Now finish problems 2 and 3a from the exponential function AND do problems 1 and 3b from the Log function || Rules for logs - converting log expressions to exponential expressions addition and subtraction become multiplication and division with logs log (a*b) = log a + log b log (a/b) = log a - log b  log a media type="custom" key="24710282" = b log a  Since there is an equivalence between logarithms and exponentials, the rules have an equivalence as well. Here is a summary of the rules, which you should know ... [|log/exp rules] Review of Friday's homework - where are the intercepts and asymptote for the Logarithmic function? can you sketch the same function using transformations? ||   || Handout of problem solving using y = C*r ^ t || Finish p 475 # 4,6,9,10 plus 4 question on the handout Only use the general formula || We worked through the example of a dollar invested at 100%/a for one year. If we earn interest annually, we will have $2 by the end of the year However, if we use compounding we can calculate the following: - semi-annual interest gives $2.25 after compounding twice in a year - quarterly interest gives $2.44 after compounding four times in a year - monthly interest gives $2.61 after compounding 12 times in a year - what would daily interest give us after a year? - what if we took this calculation to its extreme - compounding an infinite number of times in a year for an infinitely small time period? Log rules and equations - solving algebraic problems on p 491 using Log rules Answers to application problems from the handout: a.19yrs b.0.76g c.47105, 53yrs, 35yrs d.4770, 4.97hrs, e.6.65yrs, 1.49g, f.88647, 18.8 + 29 yrs, g.2.43yrs, 5.64yrs, h.25%, i.18.4yrs || answers to question c depend on the number of decimal places carried in your calculation
 * Nov 21 || Work period (mostly) for finishing Ex 10.8 trig identities PLUS the identities from the textbook that use double and compound angle formulas. || Finish trig identities from
 * Nov 22 || Today we use the chromebooks again to gather information for our next summative assignment. Even if you don't plan to hand in that assignment, you will benefit from gathering the data and determining the trig function that best fits the data. You need to gather data on daylight hours and average temperature for any city that is not in North America and that is not between 20 degrees N and 20 degrees S latitude. This way you will get variation on both graphs, since a city near the equator has 12 hrs of daylight year-round and temperatures that are consistent throughout the year. Take for example, Singapore - which is very near the equator. Your graphs will not look nearly as impressive as those for Reykjavik. And we want impressive graphs.
 * Nov 25 || Today is a day to help many students who missed classes last week. Three specific tasks that you should be (nearly) completed are:
 * Nov 26 || Finish up trig identities - do number 22, which you were told to avoid at first.
 * Nov 27 || Review of trig identities - take up number 18, which some of you found to be the hardest one.
 * Nov 28 || Work period for solving linear and quadratic trig equations - first go through trig identity # 18 and trigEquations2 # 3
 * 1) 6-19 - note that the answers also give the General Solution, which is not expected in this course. ||
 * Nov 29 || Another way to look at the angle addition and subtraction formulas.
 * Dec 2 || Quiz on Trig Identities and Equations due today - to be returned Tomorrow. Test on Wednesday
 * 2, 3a ||
 * Dec 3 || Last day for Trig Identities and Equations - review of questions from homework - according to many of you, this is the hardest unit so far. Practice the key types of identities and equation solving, and you will be fine. ||  ||
 * Dec 4 || Block B - wrapup for adopt-a-family
 * Dec 5 || Grade 8 day - no class ||  ||
 * Dec 6 || Homework - most of you got stuck in the same spot - finding the inverse of an exponential function.
 * Dec 9 || Review of the two main rules for working with Logs - we use these a lot when evaluating log expressions or when solving for exponential equations. There are three other rules for logs - just as there are three (main) rules for exponentials:
 * Dec 10 || Introduction to exponential growth and decay - this is a continuation of what you did in grade 11. However now we use one formula for all growth and decay problems, rather than using separate formulas for compound interest, exponential growth and exponential decay.
 * Dec 11 || Introduction to continuous growth y = Ce^(r*t) - [[file:guetter/RichMan.pdf|story of a rich man]]

homework: finish the handout - only use y = C r media type="custom" key="24710306", not the formula with e

p 491 # 7

|| - finish the recent work on evaluating logs, log rules, and log equations (466, 475, 491) - finish the handout on problem solving with the general exponential function - expGrowthDecay - finish the quiz for Monday - I still plan to have the unit test on Wednesday next week, so that we get it out of the way before the holidays. There are a few items remaining, but we will take care of them later next week or during the first week of January: - exponential equations - continuous growth problems - working with e media type="custom" key="24710292" and ln(x) || quiz distributed in class asks you to solve problems using the general exponential function (continual, not continuous growth/decay) || learn about the continuous growth model y = C e media type="custom" key="24758240" review for test tomorrow ||  || Additional time spent on solving applications with the continuous growth function ||  || Applications of log - earthquakes, pH, and decibel scale pH Level = -Log(concentration of hydrogen ions) pH = -Log(H+) Earthquake intensity = Richter magnitude = exponent with base 10. 6.7 magnitude has intensity/energy of 10 media type="custom" key="24758316"
 * Dec 12-13 || For those who are gone on Friday, make sure you take of the following:
 * Dec 16 || Hand in Quiz - return tomorrow
 * Dec 17 || Spend class reviewing the quiz and preparing for the test.
 * Dec 18 || Test - whole class ||  ||
 * Dec 19 || show that Lim x→oo (1 + 1/x)media type="custom" key="24710292" = e

Sound intensity = decibels. Decibel level divided by 10 gives the exponent with base 10. How many times more intense is a sound of 135 dB than a sound of 100 dB? Use the ratio of 10 media type="custom" key="24758322" /10 media type="custom" key="24758326" || P 500 # 4,6,7,8,10,13,15,17 || For those who have been avoiding summative work, you have one additional opportunity - [|Exponential Regression and Population Growth] - also worth 20 points ||  || If f(x) = 2x - 3 and g(x) = x² + 1, we can determine f(g(1)) by evaluating g(x) for x=1 and then f(x) for the value of g(x). In this case, f(g(1)) = 1 But what if we ask for f(g(x))? Then we get f(g(x)) = f ( x² + 1), which is 2 (x² + 1) - 3, which gives f(g(x)) = 2x² - 1 We also did this with my family tree - so now you know who the brother of the daughter of the daughter of the father of the father of Nicholas is... or, using function notation: b(d(d(f(f(Nicholas) = ...  You can amaze your friends with family tree math too! ||  || Review - part 1 - characteristics of functions (shape, domain/range, end behaviour, limits) Review - part 2 - graphing functions (intercepts/asymptotes or transformations) Review - part 3 - solving equations and inequalities Review - part 4 - applications of functions ||  || Jan 23 || Final Exam for Functions - since this is the last exam of the schedule, and we are the only ones in the gym at that time, I recommend that we start at 12:00 noon finish a bit early. We will decide this in class, but you should count on starting at 12:00. At that time I need any review assignments and summatives that have not been handed in. You will have a chance to review your final exam and preview your final grade in the course during the first week of second semester classes. ||  ||
 * January 6 || Exam Preparation handout - we will NOT be doing exam review for two weeks, but you will start with review today. Get to work! || [[file:MHF4UexamPrep.pdf|Exam Review]] ||
 * Jan 7 || For those who have not completed enough POINTS yet, there are still two opportunities. AND, I am reducing the required number of points from 100 to 90. You're welcome. The final exam review will be worth 20 points - 5 points per section. I will mark your method and solution for several randomly chosen problems in each of the four sections.
 * Jan 13 || Short lesson on composite functions - a function in which the x-value depends on another function - we refer to inner and outer functions - will come up in Calculus.
 * Jan 14 || First part of each next day is a summary by the teacher, and the rest of the time is for working on review and asking questions:
 * Thurs