MBF 3C Spiral 3 Review Topics covered in Spiral 3

S2L1 - Review of Sine Law and Cosine Law S3L2 - Applying the Sine Law and Cosine Law S3L3 - Graphing Stretches and Squishes (Parabolas) S3L4 - Properties of Parabolas S3L5 - Simple Trinomial Factoring Special Cases S3L6 - Forms of the Quadratic Function (Parabola equations) S3L7 - Exponential Growth Applications Too S3L8 - Exponential Decay S3L9 - The Amount of Compound Interest S3L10 - Compound Periods

Lesson 1 Using Sine Law and Cosine Law Use this as a guide for when to use each formula.

Solve the following triangles (next page) for the indicated value of x. Make sure that you know if

you are solving for a side or an angle.

Lesson 2 Applications of Sine Law and Cosine Law 1. A post is supported by two wires (one on each side going in opposite directions) creating an angle of 80° between the wires. The ends of the wires are 12m apart on the ground with one wire forming an angle of 40° with the ground. Find the lengths of the wires. 2. Two ships are sailing from Halifax. The Nina is sailing due east and the Pinta is sailing 43° south of east. After an hour, the Nina has travelled 115km and the Pinta has travelled 98km. How far apart are the two ships? 3. 3 friends are camping in the woods, Bert, Ernie and Elmo. They each have their own tent and the tents are set up in a Triangle. Bert and Ernie are 10m apart. The angle formed at Bert is 30°. The angle formed at Elmo is 105°. How far apart are Ernie and Elmo?

4. Two scuba divers are 20m apart below the surface of the water. They both spot a shark that is below them. The angle of depression from diver 1 to the shark is 47° and the angle of depression from diver 2 to the shark is 40°. How far are each of the divers from the shark? 5. To estimate the length of a lake, Caleb starts at one end of the lake and walks 95m. He then turns and walks on a new path, which is 120° to the direction he was first walking in, and walks 87m more until he arrives at the other end of the lake. Approximately how long is the lake? 6. Two observers are standing on shore ½ mile apart at points F and G and measure the angle to a sailboat at a point H at the same time. Angle F is 63° and angle G is 56°. Find the distance from each observer to the sailboat.

7. Jack and Jill both start at point A. They each walk in a straight line at an angle of 105° to each other. After 45 minutes Jack has walked 4.5km and Jill has walked 6km. How far apart are they? 8. Points A and B are on opposite sides of the Grand Canyon. Point C is 200 yards from A. Angle B measures 87° and angle C measures 67°. What is the distance between A and B? 9. A 4m flag pole is not standing up straight. There is a wire attached to the top of the pole and anchored in the ground. The wire is 4.17m long. The wire makes a 68° angle with the ground. What angle does the flag pole make with the wire?

Lesson 3 Graphing Stretches and Squishes

(x )y = a − h 2 + k Step 1. First graph the vertex (h, k) remember that the h value has the opposite sign as it

appears in the equation. Step 2. Use the pattern for graphing points related to the vertex..

Over 1 2 3 4 5 6

Up/Down 1 4 9 16 25 26

But whatever the number ‘a’ is, multiply the up/down pattern by that before you graph it.

Graph the following parabolas on the grid provided.

− (x )y = 2 + 6 2 − 8 (x )y = 41 − 4 2

Lesson 4 Properties of Parabolas For a regular parabola (one that hasn’t been stretched or squished) the equation looks like this

y = (x-h) 2+k

The vertex is : ______________ And the axis of symmetry is x = ________ There is a minimum value of ________ if there is a ____________ at the front There is a maximum value of ________ if there is a ___________ at the front Fill in the following table for the given parabola,

Parabola a h k opening Vertex Axis of Sym

max/min value

y = -4(x + 3)2 – 6 U or D ( , ) x =

max or min

y = x2 + 5

y = 0.5(x – 7)2

y = -x2 + 8

y = 2(x – 10)2 + 4

y = -5(x – 1)2 – 5

y = -(x + 2)2 + 3

y = (x – 6)2 - 2

y = -(x + 5)2

Lesson 5 - Simple Trinomial Factoring Special Cases

Always look for a common factor before trying to factor into two brackets.

Remember if the middle term is missing, the two bracket will be the same, just with opposite signs.

x x 42 2 − 2 − 2 x 5x 505 2 − 5 + 1

0x 0− 2 2 + 8 x 0x 3− 3 3 − 3 2 − 6 x 805 2 − 1 x 4x 687 2 − 1 − 1

x 6x 42 2 + 1 + 2 x 4x 64 2 − 4 + 9

x 8x 687 2 + 9 − 1 x 8x 4x4 3 − 6 2 + 6

6x 6xx3 + 1 2 − 3 x 0x4 2 + 4

Lesson 6 - Forms of the Quadratic Function

1. State the form of each quadratic function and then tell what information you know just from

looking at the function’s equation.

a) (x )y = 3 + 4 2 − 6 b) − .5(x )(x ) y = 0 + 7 − 5

c) x x y = 2 2 − 3 + 5 d) − (x )(x ) y = 3 + 2 + 4

e) − (x )(x ) y = 21 + 7 + 3 f) x x 0 y = 5 2 − 4 + 1

2. There were three equations in question 1 that were in factored form. Find the vertex of each and then state the equation in vertex form and state the axis of symmetry and max/min value.

Lesson 7&8 - Exponential Growth & Decay Applications

Sometimes you can be given in the current value and be asked to find HOW LONG it took to get from the starting value to the current value. In this case you use trial an error to find the correct

exponent. Example 1 : A rare diamond ring appreciates in value at a rate of 8% per year. How long will it take to be worth triple its value Sometimes things DECREASE exponentially as well. We call this Exponential Decay

With exponential decay, we are multiplying by something less than one. If you are given a percent of decay, you need to subtract that percent from 1. Otherwise look for words like half, or quarter that will tell you the value of b in your decay equations. Example 2. The population of a town is decreasing by 2.5% per year. If there was 10 000 people in 1995, how many people are there today (2018)? Practice problems. 1) Give examples of different equations that show exponential decay ?

a) b) c) 2) Since January 1992, the population of the city of Cowtonia has grown according to the mathematical model, where x is the number of years since January 1992.

5000(1.082) P = 7 n a) Explain what the numbers 75 000 and 1.082 represent in this model. b) What would the population be in 2013 if the growth continues at the same rate. c) Use this model to predict about when the population of Brownville will first reach 500 000.

3) A population of 700 beetles is growing each month at a rate of 3.4%. a) Write an equation that expresses the number of beetles B at time x. b) About how many beetles will there be in 2 years? 4) The half-life of a medication is the amount of time for half of the drug to be eliminated from the body. The half-life of Advil or ibuprofen is represented by the equation, where R is the amount of Advil remaining in the body, M is the initial dosage, and t is time in hours. a) A 200 milligram dosage of Advil is taken at 11:00 pm. How many milligrams of the medication will remain in the body at 6:00 am? b) If a 200 milligram dosage of Advil is taken how many milligrams of the medication will remain in the body 12 hours later? 5) Your new computer cost $3200 but it depreciates in value by about 21% each year. a) Write an equation that would indicate the value (V) of the computer at x years. b) How much will your computer be worth in 6 years? c) About how long will it take before your computer is worth close to zero dollars, according to your equation?

6) You invest $100,000 in an account with 1.01% interest, compounded quarterly. Assume you don’t touch the money or add money other than the earned interest. a) Write an equation that gives the amount of money, y, in the account after x years.

b) How much money will you have in the account after 10 years?

c) How much money will you have in the account after 25 years?

Lesson 9 & 10 - The Amount of Compound Interest

Compound interest isn’t always given once per year. It can be given multiple

times. Look for key words that tell you how many times you will get interest in a year.

annually semi-annually monthly weekly daily bimonthly biweekly

1 2 12 52 365 24 26

If the interest is given more than once a year, you must adjust the i and n for the formula to work for that compound period…

i = Yearly rate à # of compound periods n = years x # of compound periods

1) Your 3 year investment of $20,000 received 5.2% interested compounded semi annually. What is your total return? A = A = P (1+i)n

P = i = n = 2) You borrowed $59,000 for 2 years at 11% which was compounded annually. What total will you pay back? A = A = P (1+i)n

P = i = n = 3) Your allowance of $190 got 11% compounded monthly for 1 2/3 years. What’s it worth after the 1 2/3 years? A = A = P (1+i)n

P = i = n =

4) Your 6 1/4 year investment of $40,000 at 14% compounded quarterly is worth how much now? A = A = P (1+i)n

P = i = n = 5) You borrowed $1,690 for 5 1/2 years a at 5.7% compounded semi annually. What total will you pay back? A = A = P (1+i)n

P = i = n = 6) Your $440 gets 5.8% compounded annually for 8 years. What will your $440. be worth in 8 years? A = A = P (1+i)n

P = i = n =

7) Your $54,200 2 year car loan is at 15.1% compounded annually. What will you have paid for your car after 2 years? A = A = P (1+i)n

P = i = n = 8) You invest $55 at 10% compounded annually for 3 years. How much will your investment be worth in 3 years? A = A = P (1+i)n

P = i = n = 9) Your 8 year loan of $12,200 is at 5.3% compounded annually. How much will you have paid in total for your loan? A = A = P (1+i)n

P = i = n =