MCF 3MI

EXAM REVIEW

8:30 AM – 10:00 AM: Room 118 Section 01: Tuesday, January 28th

* A PENCIL, ERASER, SCIENTIFIC CALCULATOR AND RULER ARE REQUIRED *

Please Note: Your final mark in this course will be calculated as the better of:

MARK 1 OR MARK 2 Term Work 70% Final Assessments 100 % Final Assessments 30 %

MCF 3MI: Exam Review Day 1: Functions Definition: A function is a relation where every x-value maps to only one y-value Example 1: Is the relation a function? State the domain and range.

a) {(1, 2), (3, 2), (–1, 0), (1, 4)} b) c)

Example 2: Graph the quadratic function then state the following

vertex: axis of symmetry: max or min value: domain: range: x-intercepts: y-intercept:

Example 3: For the function f(x) = 2x2 – 1, determine a) f(–3) b) f(x + 3) The Algebra of Quadratics 1) Expand and simplify a) (3x – 1)(2x + 5) b) 2(a + 1) – 3(a + 4)2

2) Factor

a) 4a2b3 – 6a3b + 8a4b2 b) x2 – 3x – 28 c) x2 + 8xy + 16y2 d) 9x2 – 16y2

e) 8m2 + 6m – 5 f) –30x3 – 62x2y – 28xy2

MCF 3MI: Exam Review Day 2: Graphing Quadratics

Vertex Form Standard Form Factored Form

Example 1: Solve by factoring a) 0 = 2y2 – 8y b) 5 = 12x2 + 28

Example 2: Consider the quadratic Function f(x) = 2x2 – 8x – 42. Complete the chart

Factored form

x-intercepts

y-intercept

Axis of symmetry

Vertex

Max or min and its value

Domain

Range

Example 3: You throw your geography textbook off a cliff. Its flight is modelled by h(t)= –5t2 + 10t + 40

where h is the height in metres above the water below and t is the time in seconds. a) How high is the cliff?

b) When does the book hit the water below? Example 4: Determine the equation in factored form of the graph.

Quadratic Models

Example 1: Consider the function ( )214 8

2y x= + −

a) Describe the transformations b) Graph the function using transformations c) State the features vertex: _________________

axis of symmetry: ______________

max or min value: _______________

x-intercepts: _________________

y-intercept: __________________

domain: ___________________

range: ____________________

Example 2: Convert to vertex form by completing the square: f(x) = –2x2 – 20x + 4 Example 3: Solve using the quadratic formula 0 = 3x2 – 4x – 5 Remember the discriminant?

Remember for word problems, there are three types... 1) Given x, find y

2) Given y, find x

3) What is the max or min? MCF 3MI: Exam Review Day 3: Trigonometry In what situation is each formula used? Pythagorean Theorem 222 cba =+ Primary Trig Ratios SOH CAH TOA

Sine Law C

c

B

b

A

a

sinsinsin==

c

C

b

B

a

A sinsinsin ==

Cosine Law Abccba cos2222 −+= bc

acbA

2cos

222 −+=

Example 1: Find x Example 2: Find ∠�

Example 3: Solve the triangle

MCF 3MI: Exam Review Day 4: Sinusoidal Functions What makes a graph periodic? sinusoidal? What are the five key points of the sine function? Graph y = sin θ and list the important key features: Period:

Amplitude:

Equation of Axis:

Domain:

Range:

Transformations of y = sin θ

sin( )y a x c d= − + How does each letter affect the graph of y = sin θ? Key Features? a: c: d: Example 1: Write the equation of a sine function that has an amplitude of 0.25, a phase shift of

30o to the right and an equation of the axis at y = -3. Example 2: Graph y = 3 sin (x + 45o) + 1

Amplitude:

Period:

Equation of Axis:

Domain:

Range:

Example 3: Based on graph below showing the height of a person on a swing, determine the following: How high off the ground is the swing? What is the period of the graph? What does the period represent? Assuming the person first starts going fowards on the swing, at what height will they be after 15 seconds? Will they be on their way up/down and going foward/backward? Explain. MCF 3MI: Exam Review Day 5: Exponential Functions 1. Use exponent laws to simplify. Express answers with positive exponents. Do not evaluate.

a) 5� � �5��� 5�� b) ����� �

���� c) 3� � 9

�� � √27

2. Compute the first and second difference for each table of values and identify the relation as linear, quadratic or exponential. If it is exponential, state the value of the growth/decay rate.

x y

–2 4

–1 3

0 4

1 7

2 12

3. Consider the exponential function ��� � 3�

a) Is this exponential growth or decay? Explain. b) State the y-intercept:

Equation of the asymptote:

Domain:

Range:

c) Graph the function on the grid provided.

x y

–2 15

–1 11

0 9

1 8

2 7.5

4. There are 3400 bacteria in a culture that is growing at a rate of 5% per hour. a) Write an equation to model this growth. b) Determine the population after 1 day. c) What was the population 3 hours ago? d) When will the population reach 20,000?

5. How would the equation from # 4 change if the culture was decaying by 5% per hour?