mhf4u+old

**MHF4U1 Homework - Fall, 2012**
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 || || 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 a. y = -2x² - 8x + 10 b. y = 1/(x+4) - 3 c. y = -2 √ x - 1 + 6 || Test next Wed on Gr 11review ||
 * **Today's Date** || =**Class_Topics_Notes_Summary_and_Examples**= || **Handouts_and_**** Homework ** ||
 * Sept 4 || 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%, 3 or 4 assignments worth 15%, final exam worth 30%), the importance and value of homework and quizzes. Attendance policy.
 * Sept 5 || 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 y = a f(x-p) + q.
 * Sept 6 || Review of previous work - completing the square when A is not "1"; factoring expressions like 2x² - 5x - 3 using grouping or using Mr Limmer's handy-dandy method.

Review of last year - finding the inverse of a function. Switch x and y, solve for y, call the result f -¹ (f-inverse) 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? y = -2 (x + 3)(x - 1) y = 1 / (x-3) + 2 y = - √ x + 3 - 4 ||  || Best format for domain/range: {x € R | x > -4} {y € R | y media type="custom" key="20765744"3}
 * Sept 7 || Equations of Asymptotes - x = #, y = #

Review of inverses - how to determine the inverse of a function - by graphing and with algebra *you should be able to find inverse functions for root, quadratic, and rational functions * Finding the equation from a graph of the function - worksheet - A=1 or -1 for all functions on this sheet Take-home quiz on finding significant points and graphing with transformations - quadratic, root and rational; inverse of a function; domain and range of a function ||

|| 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 ||  || p 12 # 4,5,10; p 16 # 5 - 10 in textbook 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. || || 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 - for now you might have to ask Elijah. Rates of change - work period - finish three sections of graphing for homework ||
 * Sept 10 || Collect quiz - to return tomorrow
 * end of test material for Wednesday **
 * Sept 11 || Absolute Value
 * Sept 12 || Test on first unit - Review of Gr 11 functions - all class. If there is a chapel, you may write a few minutes into lunch ||  ||
 * Sept 13 || Review of absolute value functions. We will learn later that these are examples of COMPOSITE functions: a function within another function.
 * Sept 14 || ** 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="20845904" 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)
 * Sept 17 || Solve two problems on board with average and instantaneous rates of change

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 || || [] || finish problems on rates of change || 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 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 || p.25 # 12
 * Sept 18 || ** 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? What if x gets very close to a vertical asymptote (undefined region)? And we will learn new terminology and notation to deal with limits.
 * Sept 19 || ** End behaviour, symmetry, increasing/decreasing regions **
 * Introduction to Piecewise Functions **
 * Today we learned how to DEFINE a piecewise function - with domain. **

|| Focus on continuity: can we draw the graph without lifting our pencil from the page? are there breaks in the range of the function? 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: 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? Test NEXT WEEK THURSDAY OR FRIDAY - we will finalize this tomorrow. ||  || Finish second handout plus the review handout. p.52 # 3,4,5c, 7,9,11 ||
 * Sept 20 || Review of Piecewise functions for tax and for speeding - similarities and differences
 * Sept 21 || Review of unit - Absolute value functions, rates of change, function properties (increasing/decreasing, end behaviour, symmetry, continuity), piecewise functions.
 * Hand in quiz #2 on Tuesday** so that I can return it to you on **Wednesday** (walk-a-thon is postponed one week) || [[file:ch2ReviewMHF.pdf|ch2Review]]
 * Sept 24 || ** Test this week - I have scheduled it for Friday, but you must write on Thursday if you are not at school on Friday. Quiz due tomorrow. End of chapter problems on p 60-62 and 116-118. **

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 ||
 * Introduction to ** ** new unit on polynomial functions ** - unit outline and review of finite differences.



|| Collect quizzes - to return Wednesday 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. Do all single root graphs - we will do double and triple roots tomorrow. || || 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 || finish handouts 1.7.1 and 1.7.3 - sketch graph from factored form; find factored form from graph || - Expand factored form to get standard form - you must be able to do this without error - How to get from standard form to factored form? We want to do this because of graphing ease. - Right now, we will use long division to find out if (x+1)(x-2)(x+3) or any other terms are FACTORS of the polynomial. || Finish handout questions on finite differences - don't do text questions ||  || For those who already wrote the test, you should work on the finite differences assignment. ||  || Connection between finite differences and equation of a polynomial - how to fit data points to an "equation of best fit" - you've done this in science class. For this data: show that y = ¼x² + 2 (x,y): (2,3), (4,6), (6,11), (8,18), (10,27) Here's another example:(x, y): (-3, -15), (-1, -3), (1, 1), (3, -3), (5, -15) the solution includes A=-1 Graphs and equations of polynomials - introduction to families of functions Defn: a family of functions shares specific characteristics, which may include end behaviour, common x or y intercepts, or similar shape. Typically we talk about families of functions and how the "A" value stretches, compresses, or inverts the function; but we will expand our definition of "family" to be much more general. || p 122 # 7 p 127 # 3 p 137 # 5,7, 10-12 [|Xuru's website] || 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. example: fully factor f(x) = x⁴ - 4x³ + 3x² + 4x - 4 answer contains factors of (x-2) and (x-1) What about 6d? f(x) = 4x⁴ + 7x³ - 80x² - 21x + 270 || p. 177 # 6,7 do at least three cubics and three quartics - don't do any with degree higher than five || - if the polynomial is missing a term, insert a zero for that term as a placeholder - if the remainder is zero, the divisor (term in front) is a factor of the polynomial - if f(3)=0, then x-3 is a factor of f(x) - this means that if the remainder is zero when you divide by x-2, then x-2 is a factor of the polynomial Is there an easier way to do polynomial division? YES, today we learn synthetic division. Like other synthetics, this is not really division, but it gets us the same result TEST Next Friday - Oct 12 || Ex 3.5 p. 168-170 # 5(a-c), 6(a-d), 10(a-d), 11, 12a
 * Sept 25 || Review of Polynomials - degree, max number of turning points, end behaviour
 * Sept 26 || Hand back quizzes
 * Sept 27 || Review of graphing from factored form
 * Sept 28 || Test on Unit 2 - whole class - you may start 10 minutes before the end of lunch if you want.
 * Oct 1,2 || college fair / walkathon ||  ||
 * Oct 3 || Hand back tests
 * Oct 4 || Review of finite differences, factored form of polynomials, roots, end behaviour and graphing.
 * Oct 5 || Review of factor theorem and polynomial division - things to remember:



Synthetic Division help [|Kahn Academy] video [|PurpleMath] 4 pages ||  || 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 ||
 * Oct 9 || Introduction to polynomial inequalities - when is a function above or below the x-axis (or, in harder cases, above or below another function)?





you should finish parts A and B for Wednesday's class

remember to write your 'new' factored function in ORDER - from left to right: (x+3)(x+1)(x-2)(x-3) ||  || Today we deal with polynomial inequalities involving two polynomial functions || [|Review Problem] problems from part C ex4.3 # 6 d,e; 7 a,c,e ||  ||
 * Oct 10 || Review of +/- chart for polynomial inequalities
 * Oct 11 || Review of polynomials - factored form, standard form, intercepts, factor theorem, division, graphing, end behaviour, inequalities ||  ||   ||
 * Oct 12 || Test - Polynomials || [[file:polynomialReview.pdf|Test Review]] ||  ||
 * Oct 15 || Regression - using Excel and Web2.0 tools to determine the equation of best fit - to prepare for summative assignment #1

Introduction to polynomial regression - how can we determine the polynomial that has the best fit for a set of x-y data? We can sketch the data and then try to determine the equation (really hard); we can graph the data in Excel and add a trendline; we can use an easy, online regression utility. Here's the link:[] We will use it to find the best-fit polynomial. If the RSS value (a measure of the distance from each point to the curve) is zero, we have an exact fit. Work period for Intro Regression Handout and Summative Assignment #1 || ||  || 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 # 2,3, 5, 6 for 5 and 6 do not verify by using a graphing program. Find the intercepts and make a sketch of the reciprocal function || 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) || write the following functions as transformations, then sketch and verify the intercepts and end behaviour with calculations: y = (3x+5)/(x+2) y = (2x-6)/(x-4) y = (1-x)/(x-3) y = (4x+5)/(x+1) Do the same with p 272 # 1, 4a-c, 5a-c ||  || 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 ||  ||   || If you are given intercepts and vertical asymptotes, can you sketch the graph and determine the function? 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 16 || New Unit - Rational Functions - any function that contains a variable in the denominator. We will start with rationals that are linear in both numerator and denominator, but we will move quickly to functions that are quadratic in both numerator and denominator.
 * Oct 17 || Review: sketching rational functions where both numerator and denominator have x terms:
 * Oct 18 || work period in 120 for summative assignment - due next Tuesday ||  ||   ||
 * Oct 19 || 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?
 * Oct 22 || Review of rational functions of the form (ax+b)/(cx²+dx+e)

Review of +/- method to determining the limit of the function near the vertical asymptotes; we look at the value of the function as it approaches the vertical asymptote from both the negative side AND the positive side (and all we look for is whether the answer is positive or negative).

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? || homework: 1. y = (x+1)/(x-4) 2. y = (-2x+3)/(x+2) 3. y = (x+4)/[(x+2)(x-1)] 4. y=-(x+2)/[(x+1)(x+3)]

part 2 - p 273 #6 a,b,d sketch a graph for each and determine the function

part c find the function that best fits this data: - x-int = 3, y-int=pos, v.a. at -4,1 - x-int = 3, y-int neg, v.a. at 1,4 ||  ||
 * Oct 23 || Summative Assignment # 1 due date - not handing this in by tomorrow's class will affect your midterm grade.

Review of end behaviour/horizontal asymptotes from last class. Is there a way to do this without doing all those calculations? Hmmm... Introduction to notion of "Dominant Terms" for horizontal asymptotes = ratio of largest terms in numerator and denominator; not just the coefficients, but also the degree of x.  Rational functions in which BOTH numerator and denominator are quadratic - introductory sheet - you already have this one (rationalIntro.pdf) - you should be able to produce a good sketch for each problem on the handout - find intercepts, asymptotes, and limits, using +/- chart. Goal is to be able to predict the shape and the limits without calculating EVERY SINGLE TIME - handout of problems and graphs to sketch || ||   || Sketching rational functions without use of limits - quick sketch method. Really useful, even though I will make you use the limit method on tests and exams. Applications of rational functions - in most cases we start by sketching the part of the graph in the first quadrant (since the functions are time-dependent, and t >= 0) ||
 * Oct 24 || Quiz - rational functions



||  || - 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 Inequalities with rational functions: - 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. || ||   || Review problem - for f(x) = 2x/(x-2) and g(x) = x-3: - write f(x) in 'transformation' form and sketch it - sketch g(x) on the same - solve for f(x) >= g(x) Finish problems 3-6 from inequalities handout. Review of rational inequalities - for test on Fri 1. find intercepts, asymptotes, and limits; then sketch 2. from a graph or specific conditions, determine the equation of a rational function 3. application of rational functions - graph, find rate of change, find end behaviour 4. inequalities - when is one function above or below another (max of 2 solutions) Review of Applications || If you need more practice, work through examples 1-3 in section 5.5; also p 296 #4 b,c 5 a,b,e ||  || - what does the graph of a rational function look like? can I find intercepts, asymptotes and limits? can I correctly use the +/- chart? If I know the graph, can I determine the equation of the rational function? - if I know what a graph of a linear or quadratic function looks like, can I sketch the reciprocal function? what can I say about the reciprocal based on the original? - can I solve applications of rational functions that involve rates of change, solving for specific x and y values, looking at values near the asymptotes - given two functions, where at least one is a rational, when is one function greater than or less than the other? can I solve the equation, can I simplify and factor the inequality, can I complete the +/- chart and state the result in proper form? || Here are some additional Problems from the textbook: 255 # 2 a,c 8 c,d 10 273 # 4 a,b 5b (make sure you can do this for quadratic rationals as well - like # 14 f(x) and m(x)) 285 # 2 c 6 f 296 # 4 b, c 5 b  Apps - here are more p 274#9,10 p 287#12 p.305 #12 p 309#9,11,13 ||  || 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. 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, 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. || Sketch the following and verify a known point on the graph with the function: y= 2 sinx - 1 y= 2 sin 3x - 2 y= 2 sin(x-90) y= 3 sin 2(x-90) -1 y= -2 cos 3x + 2 y= -2 cos ½x y= -2 cos ½(x+ 180) y= -2cos½(x+ 180)+1 ||  || Ex:Temperature varies between -4 at 2 am and +8 at 2 pm. Sketch one cycle of this, starting at 2 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) ||
 * Oct 30 || Review Applications of Rational Functions - from last timekeep in mind that:
 * Oct 31 || Last chance to hand in your quiz - I only received one yesterday!
 * Nov 1 || Work period for review and test preparation
 * Nov 2 || Test - whole class - you may start right away ||  ||   ||
 * Nov 5 || 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 || [[file:guetter/MHFTrigIntro.pdf|MHFTrigIntro]] ||  ||
 * Nov 6 || 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?
 * Nov 7 || Goal of this class - move from transformations to applications.



Application: tide has a min height of 2 m at midnight, and a max height of 8 m six hours later. Sketch this function for a 24 hour period, starting at midnight, and find the function that models this behaviour. Verify a non-zero point on the graph ||  || new: introduction to radians. 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. || Homework:
 * Nov 8 || Finish trig degrees handout - the goal is to do application problems, but the steps along the way include: solving trig ratios, sketching graphs of trig functions, and solving trig equations.

p 321 # 2,7,8 (only do the positive ones) p 330 # 5 (only solve the primary trig ratios - sin, cos, tan) ||  || Focus on finding several functions that model the same phenomenon (both sin and cos), as well as solving for specific x-values (ie: at what times is the height of the tide 7 metres; for how long is the ferris wheel rider higher than 12 metres off the ground). We use sin or cos to find the first value and then we use symmetry to find the second value. Introduction to radians - fill in the handout with radians and degrees - from now on we will use only radians. ||  ||   || Review of radians - 2  ¶ rads in a circle, you should be able to write any degree measure in radians (using a fraction, where possible). ex: 300 degrees = 5 ¶/6 rad. 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 an arc length of 9¶ metres per second (or 102 km/hr). || Finish the first page of the handout Then do three problems from the 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 Ø = arcLength / radius || Finish the problems on the radian applications handout. ||  || Solving application problems using radians || soving  using radians ||   || - take up problems from yesterday's handout - new problem related to blood pressure - second side of ferris wheel handout - rates of change problems Also there are TWO quizzes that you can do in preparation for NEXT FRIDAY'S TEST. If you want good feedback, you need to hand in one or both quizzes by the end of the day on Tuesday. || ||   || Introduction to summative assignment # 2 - using trig functions to model real-life events. Your task is to choose an international, coastal city (not in the USA or Canada), and use trig functions to model the behaviour of tides, temperature, and daylight hours. You will organize your data on a spreadsheet and then use FooPlot to graph the data AND determine the function that best fits the data - you will be shown much of this in class today. || data links: [|tides] [|temperature] [|daylight hours] ||  || min temp of -2 degrees @ 2 am; max of 10 degrees @ 2 pm 1. sketch a graph for a 24 hour period, starting @ midnight 2. find both a sine and a cosine function that model this behaviour 3. for how many hours is the temperature 8 degrees or higher? Finish work on applications - textbook problems
 * Nov 12 || Review of applications with degrees - ferris wheel, tides, temperatures, etc.
 * Nov 13 || What is a radian? a segment of a circle in which the radius and the length of the arc along the circumference are equal. So, if r = 10 cm and the length along the circumference = 30 cm, that segment has an angle of 3 radians.
 * Nov 14 || 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.
 * Nov 15 || Solving trig equations using radians
 * Nov 16 || Problem solving - radian applications and periodic functions
 * Nov 19 || Turn in your quizzes by the end of class tomorrow.
 * Nov 20 || In-class quiz: Temperature problem.

- 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 1. sin² Ø + cos² Ø = 1 2. tanØ = sinØ/cosØ || p 322 # 11,12,13,16 p 345 # 9,10,11,14,16 p 361 # 6,7,8,9,10 p 371 # 7,9,12 ||  || Relationship to graphs of primary trig functions secØ = 1/cosØ = hyp/adj cscØ = 1/sinØ = hyp/opp cotØ = 1/tanØ = adj/opp || Ex 10.8 # 4,5,7,8, 9,11,12,13,15 ||  || Work on trig identities || Handout - Ex 10.8 - section B (do not try # 18 or 22 until you have finished everything else) ||   || you should come to class with a coastal city already picked, so that you are able to work with the data during this time. ||  ||   || 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 || Ex 7.2 # 5,6, 8a,c,d,e 11,12
 * Introduction to Trig Identities **
 * Nov 21 || New: The reciprocal functions - why learn them??
 * Nov 22 ||  ||   ||   ||
 * Nov 23 || Today is test day. Students gone for OFSAA championships must take this test by the end of lunch next Wednesday. If that doesn't seem fair enough, you may also write it before you leave on Wednesday. ||  ||   ||
 * Nov 26 || Form groups for adopt-a-family
 * Nov 27 || Work Period for [[file:summative2.pdf|Summative Assignment # 2]] - room 120
 * Nov 28 || Addition and Subtraction formulas

||  || You are creating a one-page summary for each of the following: tide data, daylight hours, and temperature. Each page will include, but not be limited to, a data table, a fooplot graph showing both the data and the best trig function that fits the data, the algebraic work that accompanies your graph, and an explanation of what you did, how well the data matches the function, and so on. ||  ||   || Application of new trig formulas to trig identities - proving identities with expressions like: sin(¶/2 +x) cos (2x) First we need to use the appropriate formula for the compound or double angle expression and then we need to solve the L.S and R.S. of the identity. || Ex 7.3 # 2 Ex 7.4 # 10e,f 11a ||  || When using the compound angle formulas, we are often given specific values - for example sin(¶/2 +x). It is important to remember that sin ¶/2 = 1 and cos ¶/2 = 0 This is the last day we will spend on trig identities. ||  ||   || p 436 # 6,7 ||   || 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 trig identities new: p 401 # 9,10 p 407 # 4-7 ||  || For the test you will be given the two basic trig identities, the reciprocal functions, and the compound and double angle formulas. You will be expected to: - solve trig identities using basic trig identities, reciprocal functions and double/compound angle formulas - evaluate expressions involving compound angles (Dec 3,5) - solve trig equations in which there will be at least two, but usually more solutions || ||   || Begin LAST UNIT - Exponential Functions 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. Handout of problem solving using A = C*r ^ t || ||   || For those not writing the test, you will work on the exponential applications handout. If you finish, I will have several problems from the textbook. ||  ||   || If you wrote the test yesterday, this is the time to work on the exponential applications handout and the textbook questions. Answers to application problems: 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 ||  ||   || 1. use transformations of base exp or log functions 2. use algebra to solve for intercepts, asymptotes, and end behaviour || do question 2a-e, then compare the graph based on the points and asymptotes to the graph you would get by using transformations ||  || - what if the function was y = 3 media type="custom" key="21774732" - 9? - what about y = 3 media type="custom" key="21774738" - 6? why is this one harder? what Log operations do you remember? here is the most important one for our work today: log media type="custom" key="21774694" c = log c / log a - now try y = 2 media type="custom" key="21774744" - 4 and y = 2 media type="custom" key="21774748" - 5 || Finish problem 2 on handout; then do # 3a Then, find the inverse and sketch both the function and the inverse for: y = 2 media type="custom" key="21774754" + 8 y = 4 media type="custom" key="21774760" -5 y = 3 media type="custom" key="21774764" - 12 ||  || finding intercepts for log functions using rules for logs || handout # 1,3b (Dec 14) cross out the vertical stretch of 3 in each. ||  || 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="21796100" = 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] || ||  || Finding the inverse for a log or exponential function - Dec 14 worksheet Introduction to continuous growth y = Ce r*t - 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? We would have, which has an approximate value of 2.7182818. This tells us that the result of continuous compounding (like in bacteria and population growth and radiation problems) is better modeled by the function y = emedia type="custom" key="21809266" Solve problems from earlier using the continuous growth model: y = c e r*t. We need to use Loge to solve this type of problem. Log eis used quite often in advanced mathematics, so it is given a special name and formula: we call it the Natural Log and the formula is ln, pronounced LON (or lawn, if you want). The equivalent expressions are:Log media type="custom" key="21873502" x = ln(x) || Here's a sample problem we did in class. A bacteria culture grows from 1000 to 5000 in 20 minutes. Using a continuous growth model, determine the number of bacteria after one hour.
 * Nov 29 || Work Period for Summative Assignment # 2 - room 120
 * Nov 30 || Double angle formulas - extension of the addition and subtraction formulas
 * Dec 3 || Review compound and double angle formulas by taking up 7.4 3 10f and 11a
 * Dec 4 || Work period on trig identities - specific focus on identities with compound and double angle formulas in them || P 427 # 8
 * Dec 5 || Another way to look at the angle addition and subtraction formulas.
 * Dec 7 || Review of trig identities and trig equations for Tuesday's test.
 * Dec 10 || Anything to take up from Trig Equations?
 * Dec 11 || Test - block B is right after lunch, so you may start anytime in the last 15 min of lunch
 * Dec 12 || Alternate test time - mostly for those in the play, but not limited to them.
 * Dec 13 || Adopt-a-family wrap-up time ||  ||   ||
 * Dec 14 || Graphing exponential and logarithmic functions - 2 ways
 * Dec 17 || Review of y = 3 media type="custom" key="21774726" - 3 where are the intercepts and asymptote? can you sketch the same function using transformations?
 * Dec 18 || graphing log functions based on transformations
 * Dec 19 || Rules for logs - converting log expressions to exponential expressions
 * Dec 20 || Exponential and Logarithm Test on Thursday, Jan 10.

A = Ce r*t where A = 5000, C = 1000, t = 20 you need to use the natural log to solve for r = 0.08047 Here are a few steps: 5000 = 1000e r*20 ln(5) = ln( e r*20) and since ln and e are inverses, we are left with 1.609 = 20*r which you can solve for "r". And then you can solve for the number after one hour, which is 125,000

Now solve this one: Use the continuous growth model to find the population of a city in 2020 if it had 35000 in 1975 and 85000 in 2000. my answer is 172, 860. ||  || Test on exponential and logarithmic function this Thursday - right after lunch. Review of exponential and logarithmic functions, starting with the last topic - continuous exponential growth. We use a continuous growth model because it best represents how a population grows or decays. There are many similarities between the two growth models, which we will look at. Quiz - I can give you feedback if you hand in your quiz by the end of tomorrow's class. || ||   || - graphs, transformations, intercepts, asymptotes, limits, rates of change for exp/log functions - simplifying and solving equations using the laws of exponents and logarithms - applications of exponential functions - both standard growth model and continuous growth model Introduction to applications of logarithms - sound intensity, earthquakes, and pH level (from p 500) || quiz due by end of school day begin p 500 # 4,6,7,8,10,13,15,17 Log applications are NOT on this unit test, but may show up on the exam... ||  || Earthquake intensity = Richter magnitude = exponent with base 10. 6.7 magnitude has intensity/energy of 106.7 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 1013.5/1010 || p 500 # 4,6,7,8,10,13,15,17 this is NOT on tomorrow's test
 * Jan 7 || Exam Review handout - you need to review almost everything; this is just a starting point for your work.
 * Jan 8 || Review of all material from exponentials and logarithms
 * Jan 9 || pH Level = -Log(concentration of hydrogen ions) pH = -Log(H+)

Here is extra practice from ch 8 p 510 # 7,8,10,11,13,14,15 p 512 # 6,7,8 ||  || pH Level = -Log(concentration of hydrogen ions) pH = -Log(H+) Earthquake intensity = Richter magnitude = exponent with base 10. A 6.7 magnitude has intensity/energy of 106.7 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 1013.5/1010 || p 500 # 4,6,7,8,10,13,15,17 ||  || Review - part 2 - graphing functions (intercepts/asymptotes or transformations) Review - part 3 - solving equations and inequalities Review - part 4 - applications of functions First half of each class is formal teaching/review; second half is to work through review exercises. Friday's class is a Q&A session. ||
 * Jan 10 || Test - whole period - you may start 10 min before the end of lunch. ||  ||   ||
 * Jan 11 || New: applications of Log - sound intensity, earthquakes, and pH level (examples from p 494 - 498)
 * Jan 14 - 18 || Review - part 1 - characteristics of functions (shape, domain/range, end behaviour, limits)

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 * Jan 21 || MHF4U Final Exam - 9:30 AM ||  ||   ||