Foundations of Math 12

Final Exam Review Tips for Studying:

1. Answer each question in this review booklet 2. Rewrite your notes

3. Attend each lunch time tutorial your teacher holds (FOM12 is Tuesday and Friday afternoons) 4. Make cue cards and quiz yourself

5. Make a 6 window for the entire course 6. Look at the learning goals that are worth the most marks and MASTER those sections

7. Find a study group that BRINGS OUT THE BEST IN YOUR LEARNING 8. Teach your parents, relatives, siblings, or pets this information

9. Go to my website (www.beckersciences.weebly.com) and review the videos and powerpoints

GOOD LUCK ☺

Learning Goals Final Exam Review Booklet Be

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1. I can describe characteristics of polynomial graphs

2. I can use technology to perform lines/curves of regression

3. I can graph a sinusoidal function

4. I can use technology to perform a sinusoidal regression

5. I can graph exponential and logarithmic functions

6. I can use technology to perform an exponential and logarithmic regression

7. I can use Venn Diagrams to solve problems

8. I can solve probability situations

9. I can solve permutations and combinations

10. I can explore risks of credit card use

11. I can calculate mortgage options and assess risks involved

12. I can calculate changing interest rates and payment options

Name:

___________________________________

______

Learning Goals #1: I can describe characteristics of polynomial functions What is the degree of a linear function?

What is the degree of this function?

Describe the end behaviour of the function f(x) = –5x2 + x – 2.

Describe the end behaviour and intercepts (both x and y) of the constant function f(x) = 10.

What is the maximum number of turning points for the below function? f(x) = –5x3 + x2 – 2.

Determine the independent and dependent variables for the following relationship: The depth of the tide is related to the hours after midnight. Independent: _______________________________ Dependent: _________________________________

Graph the below function: f(x) = (x + 3)2 – 4

Graph the below function:

f(x) = −3

4𝑥 + 2

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

The volume of water in a cylindrical water tank being drained can be modelled by the function v(t) = 3.8t + 225 where v represents the volume in litres and t represents the time in minutes. How much water is left in the table after half an hour?

Graph the following polynomial function by hand and state the following: y = (x + 1)3 - 2

a) Number of turns b) Degree of polynomial c) Number of x-intercepts d) Co-ordinates of y-intercepts e) End behaviour f) Domain g) Range

Circle which graphs are functions:

Learning Goals #2: I can use technology to perform lines/curve of regressions (use DESMOS) A horse jumps over a course obstacle. The horses’s height, in meters, above the ground, is given in the table below.

Using quadratic regression, estimate the maximum height of the horse during the jump.

Height (m) 0.7 1.80 2.56 0.99 0.23

Time (s) 0.5 1.0 1.5 2.5 3.5

The cost of a taxi ride in Summerland, for several distances, is given in the table below.

Distance (km) Cost ($)

40 70.49

2 8.40

90 152.77

5 10.89

20 37.76

35 64.16

21 40.93

12 26.55

28 52.09

a) Use DESMOS to create a scatterplot, and describe the relationship that you observe b) Determine the equation of the linear regression function that models the data c) Use your equation to estimate…

a. The cost of a 50 km trip b. The fixed cost for any taxi trip

Determine the linear regression for the data

x 1 2 3 4 5 6

y 84 155 241 310 405 478

What is the equation?

What is the r value and what does this mean?

The growth of a tree can be modelled by the function H(t) = 2.3 t – 0.45 Where h represents the height in metres and t represents the time in years. Approximately how long will it take the tree to grow 32 m tall?

Learning Goal #3: I can graph a sinusoidal function

Graph the following polynomial function by hand and state the following: f(x) = sin 𝒙 + 𝟏 a) Amplitude: ___________ b) Period: _____________ c) Equation of midline: ___________ d) Domain: ______________ e) Range: ____________

What is the equation for the below sinusoidal graph?

f(x) = ________________________________

What is the equation for the below sinusoidal graph? f(x) = ________________________________

Determine the amplitude of the following function.

y = 5 sin 1.5(x + 60°) – 5

Determine the period of the following graph.

What is the equation of the midline of y = cos x?

Determine the midline of the following graph.

Learning Goal #4: I can use technology to perform a sinusoidal regression (use DESMOS)

A seat’s position on a Ferris wheel can be modelled

by the function

y = 18 cos 2.8(x + 1.2) + 21,

where y represents the height in feet and x represents

the time in minutes.

Determine the diameter of the Ferris wheel.

The height of a mass attached to a spring can be

modelled by the sinusoidal function

h(t) = 53.5 – 4.2 cos 23.5t

where h(t) represents the height in centimetres and t

represents the time in seconds.

What is the height of the mass, to the nearest tenth

of a centimetre, after the first minute?

The average depth of the water at an ocean port can be modelled by the function

h(t) = 0.76 cos (0.25t) + 3.82

where h(t) represents the depth in metres and t represents the time in hours after 5:00 p.m. on April 19,

2012.

a) What is the minimum depth of water, to the nearest centimetre? Show your work.

b) Estimate the depth of the water at 10:30 a.m. on April 20, 2012. Show your work.

c)

The following table gives the time of sunrise recorded on the first of the month in a British Columbia town.

Month Jan. 1 Feb. 1 Mar. 1 Apr. 1 May 1 Jun. 1

Sunrise 07:28 07:01 06:34 06:10 05:47 05:38

Month Jul. 1 Aug. 1 Sep. 1 Oct. 1 Nov. 1 Dec. 1

Sunrise 05:44 05:58 06:27 06:58 07:26 07:45

Use sinusoidal regression to determine the earliest and latest sunrises possible in this town. Round values to

the nearest minute. Show your work.

Learning Goal #5: I can graph exponential and logarithmic functions How many x-intercepts does the logarithmic function

f(x) = log4 x have?

How many y-intercepts does the logarithmic

function f(x) = log4 x have?

Determine if the exponential function f(x) = 1

2

𝑥 is

increasing or decreasing, explain how you know.

How many x-intercepts does the logarithmic

function f(x) = 2x have?

Determine the x and y-intercept of the exponential

function f(x) = -2(4𝑥) x-intercept: __________________________ y-intercept: __________________________

What is the domain and range of the logarithmic

function f(x) = 2log3 x?

Domain: __________________________

Range: ___________________________

Graph the following function:

y = (2)x

Graph the following function:

y = log2 (x)

State the below characteristics:

Function equation: ___________________________

x-intercept: ___________________________

y-intercept: ___________________________

asymptote: ___________________________

domain: _____________________________

range: _______________________________

end behaviour: ________________________

State the below characteristics:

Function equation:

___________________________

x-intercept: ___________________________

y-intercept: ___________________________

asymptote: ___________________________

domain: _____________________________

range: _______________________________

end behaviour: _________________________

Learning Goal #6: I can use technology to perform an exponential and logarithmic regression (use DESMOS)

Jack accidently inhaled poisonous fumes. He was transported to the hospital, arriving four hours after the

incident. At the hospital, he was given treatment and the concentration of poison in his blood was

measured every two hours. The table below shows the measurements recorded since the time of the

incident.

Concentration (µg/cm3) 33.4 27.5 23.3 19.8 17.2

Hours Since Accident (hr) 4 6 8 10 12

The data in the table can be modelled by a logarithmic function of the form y = a + b(ln x), where y is the

number of hours since the accident and x is the concentration of poison in the blood in µg/cm3

Jack is released from hospital when the concentration of poison in his blood is less that 2 µg/cm3. Using the

regression equation, how long did he spend in hospital?

Colleen wants to invest money for her post-secondary funds. The number of years, y, that it takes for an investment of $2000 to increase in value to x dollars can be modelled by a logarithmic function of the form y = a + b ln x. Below is a table that outlines Colleen’s investment over a period of 10 years.

What is the equation that models this investment gain? How many years will it take to double Colleen’s original investment? What is the value after 15 years?

Value of Investment, x

Number of years, y

2088 2

2190 4

2245 6

2598 8

2790 10

Learning Goal #7: I can use Venn Diagrams to solve problems

Katherine surveys the students in her class with respect to their post-secondary goals. She records the results in a Venn diagram, as shown below. There are 30 students in the class.

Each • represents 1 person in the class.

Write three probability statements that have 3

30 as the answer.

In a survey of 55 people:

• 14 people like Hawaiian pizza. • 20 people like pepperoni pizza. • 25 people like cheese pizza. • 15 people do not like pizza. • 5 people like Hawaiian pizza and pepperoni pizza, but not cheese pizza. • 1 person likes all types of pizza. • 2 people like Hawaiian pizza and cheese pizza, but not pepperoni pizza.

How many people like only cheese pizza? Draw a Venn diagram to show your process for your solution

Learning Goal #8: I can solve probability situations

In biological study on genetically modified mice, 45% have blue eyes, 30% have a short tail and 20% have both blue eyes and a short tail. What is the probability that a randomly selected mouse from this study has neither blue eyes nor a short tail?

The probability that Katie will make a free throw in basketball is 0.53. The probability that Ashley will make a free throw is 0.33. Assume independence. What is the probability that at least one of them will make a free throw on their next shot?

The calendar for the month of February is shown below. Cassidy’s birthday is in February.

What is the probability that her birthday is on Valentine’s Day (February 14) or on a weekend (Saturday or Sunday)?

A soccer team has practice jerseys in three different colours. The team bag contains 4 yellow, 6 white, and 5 orange jerseys. Brianna randomly gives Danielle and Stubbs each a jersey. Which expression correctly represents the probability that both jerseys are the same colour?

Learning Goal #9: I can solve permutations and combinations

Rogers Arena has 7 gates. In how many ways can you enter the arena and leave the arena by a different gate?

In Holland, license plates start with any two digits followed by any three letters except A, E, I, O, U, C and Q, followed by a single digit.

If no letters are repeated, how many different license plates are possible with this configuration?

A leadership class has 25 students. How many ways can a school chairperson, secretary and treasurer be selected from the class?

Susan is playing a game of Scrabble. She picks the following 7 tiles from the bag.

In how many ways can she arrange all 7 tiles on her tray?

There are 18 boys and 13 girls in an English class. A group of 6 students is needed to read from a play. If there are 2 roles for boys, 3 roles for girls, and a narrator who could be a boy or a girl, how many different groups of 6 students are possible?

Learning Goal #10: I can explore risks of credit card use

At the start of April, Rhys decides to purchase a bicycle for $2000 (including taxes).

• He pays $1000 cash as a down payment and pays the balance using his credit card.

• He pays $500 towards his credit card balance at the end of the month.

• He pays the remaining credit card balance at the end of the following month.

• Each month his credit card company charges 20% per annum, compounded daily, on the outstanding balance before each payment.

How much is due on his last payment?

Monty has a credit card balance of $5200.The credit card company charges 19.5% interest, compounded daily. Monty decides to stop using his credit card and to make monthly payments so he can pay off his debt.

a) How long will it take Monty to reduce his current credit card balance to zero if he pays $250 a month?

b) If he doubles his monthly payment to $500, how much sooner will his debt be paid off?

c) How much interest will he save if his

monthly payment is $500 rather than $250?

Stephon and Mairin both want to become debt free by paying off their credit card debts at the same time.

• Stephon has $4512.03 on his credit card that charges an interest rate of 12.4%, compounded daily. He wants to pay off his debt by making regular monthly payments of $250.

• Mairin has $3010.22 on her credit card and she wants to pay off her debt by making regular monthly payments of $170.

What annual interest rate is Mairin being charged if her debt compounds daily?

Learning Goal #11: I can calculate mortgage options and assess risks involved

A land development company purchased a rental property for $279,000. They made a 10% down payment and negotiated a five year mortgage at 6.95% amortized over 25 years.

Determine their monthly payment.

Carlos was approved for a mortgage to finance his new

house that he purchased for $325 000. He made a down

payment that was 20% of the purchase price. The

mortgage is compounded semi-annually at an interest

rate of 4.2%. Carlos will repay the mortgage in 25 with

regular monthly payments. How much will each monthly

payment be?

Vladimir is buying a house that costs $375 000. He has negotiated a mortgage with the bank that requires a down

payment of 12% of the cost of the house. He will pay off the mortgage with regular monthly payments over 25 years

at an interest rate of 2.8%, compounded semi-annually. How much will he pay in total?

Dan and Suzy are buying a house that costs $545 500 and have saved enough to put 10% down. They will pay off the mortgage with regular monthly payments over 25 years at an interest rate of 3.29%, compounded semi-annually. How much will their monthly payment be (assuming their mortgage rate stays the same for the life of the mortgage)? After 25 years, how much will they have paid for the house in total?

Learning Goal #12: I can calculate changing interest rates and payment options

How can you determine the approximate amount of time it takes for an investment to double?

Nik is talking to a car salesman about leasing an electric car. The salesman informs him:

• The lease is 4 years and interest rate is 1.0%

• The down payment is $5,000 and buyout is $12 000

• If Nik purchases the car at the end of the lease, the total cost of the car is $45 000

What is the monthly payment?

Julia graphed four curves representing the same initial value of an investment compounded annually, quarterly, monthly and weekly. Which curve represents quarterly compounding? How do you know?

Jack decides to purchase a bicycle from Island Cycle for $2000 (including taxes). He considers two options:

OPTION A OPTION B

• Buy outright with cash • Buy now, pay later

• Pay an initial administration fee of $20 in cash

• No down payment

• 8% per annum interest, compounded monthly over 1 year

How much more must he pay if he chooses Option B instead of Option A?

Alyssa borrowed $250.00 from a payday loan company and had to repay the loan plus a $25.00 fee after 15 days. Calculate the annual interest rate that fee would equate to.