MBF3C S3L1 Sine Law and Cosine Law Review May 08, 2018

Transcription

1 MBF3C S3L1 Sine Law and Cosine Law Review May 08, 2018 Topic : Review of previous spiral I remember how to apply the formulas for Sine Law and Cosine Law Review of Sine Law and Cosine Law Remember when finding an angle with Cosine Law, the subtracted side in the formula is across the triangle from the angle you are finding Cosine Law If you have a non right triangle with all three sides you can use Cosine Law to find an angle Sine Law If you have a non right triangle with an angle side pair having known values you can use Sine Law to......find a side (if you know a second angle) Find side 'a'. If you have a non right triangle with two sides and the angle between them, you can use Cosine Law to find the third side. find an angle (if you know a second side) Find angle 'A'.

2 MBF3C S3L2 Applying the Sine Law & Cosine Law (2).notebook Example 1. Topic : Applying Sine Law and Cosine Law I can set up word problems involving non right triangles and solve using Sine Law or Cosine Law. May 08, 2018 A cottage under construction is to be 12.6 m wide. The two sides of the roof are to be supported by rafters the same length that meet at a 500 angle. How long should each rafter be? Applying Sine Law and Cosine Law How can we tell when to use Sine Law and when to use Cosine Law? Think about these cases... Case 1 I have an angle and the side across from it with known values plus one other piece of information. Example 2. Case 2 I have a triangle with the side lengths of all three sides known. Case 3 I have a pacman situation (two sides with values, plus the angle between them) Case 4 I have two angles and the side between them with known values. Example 3. A hockey net is 2m wide. A player shoots from a point where the puck is 3.2 m from one goal post and 4.4 m from the other. Within what angle must he make his shot to hit the net? Practice Questions Page 42 #3 9 (notice that 3 & 4 don't actually ask you to solve the problem) Extra Application Problems Page 25 #5, 9, 10 Page 36 #4, 8 10 Ships A and B at sea are 15.6 km apart. A port (maked by C) can be seen from the deck of each ship. The angles between the line joining the ships and the line of site to the port are 580 and 720 respectively. How far is each ship from the port?

3 MBF3C S3L3 Graphing Stretches and Squishes May 08, 2018 Topic : Graphing Stretches and Squishes I know how to change the equation of a parabola so that the shape of the parabola is transformed and I can graph this new parabola Transformations of Parabolas x2 y= y x left/right Notice that for each HORIZONTAL DISTANCE you move away from the VERTEX, you move up that same distance SQUARED then multiplied by the 'a' value. This happens whether you move left or right. Example 1. State the vertex of each parabola and then graph it using the pattern we discovered. Mark on the axis of symmetry for each up 1 unit 2 units 3 units 4 units 5 units... n units What happens if the "a value" is negative? Example 2. Graph the following parabolas. Pay attention to the direction of opening. a) y = 3(x+1)2 10 b) y = 1 (x 2)2 2 a) y = 3(x 2)2+10 b) y = 1(x+1)2 4 2 c) y = 1 x d) y = 2(x+6)2 4 c) y = 1x d) y = 2x2 4 d) y = 4(x 3)2 10 e) y = 6(x 4)2 d) y = 4(x 3)2 +10 e) y = 6x2 Practice Questions Worksheet

4 MBF3C S3L3 Graphing Stretches and Squishes May 08, 2018

5 MBF3C S3L4 Properties of Parabolas May 08, 2018 Topic: Properties of Parabolas Example 1. Fill in the following chart for the properties of parabolas. I know the basic properties of parabolas and can state them just by looking at the equation of a transformed parabola. Properties of Parabolas Equation a h k Opens Vertex y = (x+5) 2 7 y = 2(x 1) Axis of Symmetry Max/Min Value There are 5 main properties of a parabola, and all can be picked out from the equation without having to graph the function... y = 1/3(x 9) 2 a, h, k Opening Vertex Axis of Symmetry Max/Min Value y = 3x y = 1/5x 2 y = 6(x+7) 2 10 a, h, k Opening Vertex Axis of Symmetry Max/Min Value Example 2. Graph the above parabolas on the grid provided. a, h, k Opening Vertex Axis of Symmetry Max/Min Value In General Opening : if 'a' is positive if 'a' is negative Vertex : Axis of Symmetry : Maximum : if it opens up Minimum : if it opens down

6 MBF3C S3L5 Simple Trinomial Factoring Special Cases May 08, 2018 Topic : Simple Trinomial Factoring Special Cases I know how to factor some quadratics that have common factors or are missing terms. Simple Trinomial Factoring Special Cases Special Case #1 there is a common factor Special Case #2 there is no x term 5x 2 45 First of all you want to remove the common factor. The number in front of x squared can be divided out of both terms. Then just know that the middle term missing means that the coefficient of the x term must have been zero. Sometimes a simple trinomial can be disguises as a complex one by a common factor. 3x 2 + 3x 18 At first it looks complicated, but once you realize that you can take out a common factor of 3, it's really a very simple trinomial to factor. 3x 2 + 3x 18 Special Case #3 there is no constant term y=4x 2 +20x First take out the common factor. But this time since the constant term is missing, you can also divide out an x along with it. So, if you see a number in front of the x 2 term, you will likely be able to divide every term in the trinomial by that number. Then you can just ignore it, and factor as usual. What are the x intercepts of this quadratic function? Examples. Factor each of the following quadratic expressions. a) 3x x + 36 b) 0.5x 2 5x + 8 c) 2x 2 10x 48 d) 4x c) 6x 2 30x

7 MBF3C S3L6 s of the Quadratic Function May 08, 2018 Topic : s of Quadratic Functions I know the three forms that a quadratic function can be written in and what information can be taken directly from the equation for each. Example 1. For y = 3x 2 + 4x 7 state... a) the direction of opening b) does it have a max or min? s of Quadratic Functions Using technology, graph each of the following functions. What do you notice? They all represent the same parabola! A. y = x 2 + 2x 3 B. y = (x + 3)(x 1) C. y = (x+1) 2 4 Standard Factored Vertex y = ax 2 +bx+c y = a(x r)(x s) y = a(x h) 2 +k c) the y intercept? y = 3x 2 + 4x 7 Example 2. a) What are the x intercepts of f(x) = (x + 6)(x 4)? b) Use the x intercepts to locate the vertex. c) What is the y intercept? Graph the parabola and state its properties. Vertex : Opening : Axis of symmetry: Max/Min Value : x intercepts : y intercepts : y = (x + 6)(x 4) Because of symmetry, the vertex will be directly between the two intercepts. So take the average of the intercept points and you will find the x coordinate of the vertex. If you know the x coordinate of the vertex, the y coordinate is simply the value of the function at that location, so determine f( 1). Which properties is each form useful in finding? Standard y = ax 2 +bx+c Factored y = a(x r)(x s) Vertex y = a(x h) 2 +k y = x 2 + 2x 3 y = (x + 3)(x 1) y = (x+1) 2 4 NOTE When the parabola goes across the y axis, the x value is ZERO. To find the y intercept, let x=0. The vertex really is the most important part of the parabola. Once you know it, you also know * axis of symmetry * max/min value * range Practice Questions Handout Page

8 MBF3C S3L7 Exponential Growth Applications Too May 08, 2018 Topic : Exponential Growth I can create equations for exponential growth situations. Exponential Growth Current Value y = Ab x Number of Time Periods Starting Value Growth Factor We are going to do a example, where finding the x value (the number of times something grows) is a little more complicated. Example 1. In 1998, there were 96 children born in Mitchell. If the number of births, since 1998 grows at a rate of 2.1%, determine the following: a) an equation to represent this situation. b) the number of children that will be born in Mitchell in 2050? Example 2. Ms. Sinclair was growing a bacterial culture with her grade 11 biology students. The culture started with 450 individual bacterium and doubles every 3 hours. Example 3. The population of a town doubles every 7 years. If there are 4500 people people in the town now, how long until it has people? a) Determine an equation to represent this situation. b) How many bacteria would be present after 4 days? Example 4. An investment grows by 12% per year. At this rate, how long will it take to earn triple its original value?

9 MBF3C S3L8 Exponential Decay Applications May 08, 2018 Topic : Exponential Decay I can create equations for exponential decay situations. Exponential Decay Exponential decay occurs when value is LOST repeatedly. The general form of the equation looks the same, but the value we multiply by everytime (b the base) will be between 0 and 1. Example 1. The Decay Rate is given as a percent. Electronics lose their value rather quickly since newer models are always coming out. It seems that as soon as you buy a new computer, it's already obsolete. A Macbook Pro, purchased for $2400, depreciates at a rate of 2% per month since it was purchased. a) Create an equation that models the value (V) of a macbook pro, n months since it's purchase date. Since it's decaying, b = 1 rate (as a decimal) When we multiply by a number between 0 and 1, our result is always smaller than we started with. Current Value y = Ab x Number of Time Periods b) What can you expect to sell the computer for, when you upgrade to a new on 3.5 years later? Starting Value Growth Factor When we talk about something decaying in value (where value is given in terms of money), the more common term we use is DEPRECIATION Practice Questions Handout #1 5, 7, 8a, 9a, 14 Example 2. Decay Rate is given as a FRACTION (or decimal). When you drink caffeine, it leaves your body according to an exponential rate of decay. It takes 4 hours to lose half the amount of caffeine in your system. We say that caffeine has a half life of 4 hours. a) Write an equation that models the percent of caffeine in your blood (P) if x is the number of times caffeine is reduced by half. b) Determine what percent of caffeine will be present in your system, 20 hours after drinking an energy drink? Practice Questions Handout #1 3, 5 10

10 MBF3C S3L9 The Amount of Compound Interest May 08, 2018 Topic : The Amount of Compound Interest What we really did, was multiply $3000, by times. A shorter way to write that is... I can use the compound interest formula to calculate the amount of an investment. The Amount of Compound Interest We saw in the last lesson that compound interest is like investing a new larger amount of money every year. It amounts to interest earned on top of interest. We are going to look at the compound interest question again, only from a slightly different angle. Example 1. Josh deposits $3000 in an account that pays 7% compound interest. How much would he have after 5 years? Every year, Josh keeps 100% of his $3000 (although he can't use it, it is still his) plus the bank pays him 7% more. At the end of each year, he has % of the principal he started with. Instead of multiplying by 0.07 and then adding on the principal, we can do this in one step by multiplying by For compound interest, since we are multiplying by the interest rate repeatedly, this is actually an application of exponential functions. The compound interest formula looks like this... Amount after n compound periods A = P (1 + i) n Principal (initial amount invested) interest rate per compound period number of times we get interest Example 2. Determine the amount if $2500 is invested at 3.5% compounded annually for 5 years. How much interest is earned? Principal at the start of the year Amount at the end of the year Year 1. Year 2. Year 3. Year 4. Year 5. Example 3. Isaac and his dear friend Marty, travelled back in time to They were stuck there for 6 months while Marty fixed the Delorean. In that time they got a job and managed to save up three month's salary ($309!) Before they departed, Isaac put in into a bank account that paid 7% compound interest. How much money was waiting for him when he got? How much interest did Isaac earn? Practice Questions Page 225 #1, 2, 7, 8, 10

11 MBF3C S3L10 Compound Periods (2).notebook May 08, 2018 Topic : Compound Periods I know the how to adjust the interest rate and number of compound periods if interest is compounded over different time periods. Compound Periods Amount after n compound periods A = P (1 + i) n Principal (initial amount invested) interest rate per compound period number of compound periods per loan NOTE Unless otherwise stated, interest rates given are annual and investment durations are given in years. The value you use for "i" and "n" must be calculated as follows... i = (annual interest rate) / (# of compound periods in a year) n = (duration in years) x (# of compound periods in a year) When you hear something like "compounded semi annually", this is telling you how often interest is given. Semi annually means twice a year. List of compounding periods... Term annually semi annually (twice a year) # of Periods a Year Example 2. What's a better investment for your $20 000? A. 7.75% compounded semi annually B. 7.24% compounded weekly monthly weekly daily quarterly bi weekly (every 2 weeks) bi monthly (twice a month) Example 1. Let's revisit Isaac's trip back in time 88 years. Suppose that the account he deposited his money in was compounded monthly over that time. (Remember it was 7% and he deposited $309). How much money would he have when he got back to the future? Practice Questions Page 232 #3, 4, 8, 9