MCR3U - Finals

Vocabulary

• Sum - The result of addition

• Difference - The result of subtraction

• Product - The result of multiplication

• Quotient - The result of division

• Term - A value that is either a coefficient or variable, or a product of multiple

• Like Terms - Terms that are exactly the same except for possible their coefficients

• Algebraic Expression - A term or terms connected with algebraic operations (adding, subtracting, multiplying, etc.) and/or grouping symbols (brackets, roots, etc.)

• Equation - A statement that the values of two expressions are equal, indicated by the symbol =

• Simplify - Combine like terms and factors. Remove brackets and grouping symbols. Reduce. Rationalize the denominator. Evaluate exactly. Write fractions as mixed numbers. Write with only positive exponents. State restrictions and variables.

• Evaluate - To solve using a certain method

• Solve - To find all the values of the variables that make the equation true

• Factor, v. - To rewrite as a product

• Factor, n. - A divisor; numbers/polynomials that are multiplied together to form an expression

• Relation - A connection/correspondance between the values of two variables

• Function - A special relation that connects each value of the independent variable to exactly one value of the dependent variable

• Domain - The set of all possible values for the independent variable

• Range - The set of all possible values for the independent variable

• Natural Numbers - N : Any number in the sequence 1, 2, 3, ... ; commonly referred to as the counting numbers

• Whole Numbers - W : Any number in the sequence 0, 1, 2, ...; all natural numbers as well as zero. They are the answers to the question "How many?"

• Integers - Z : Any number in the sequence ... -3, -2, -1, 0, 1, 2, 3, ... ; all of the whole numbers and their opposites

• Rational Numbers - Q : Any number that can be written as a ratio (fraction) of two integers; any number that when written in decimal form has a terminating or repeating decimal

• Irrational Numbers - Q' : Any number that can NOT be written as a ratio (fraction) of two integers; any number that when written in decimal form has a non-terminating and non-repeating decimal.

• Real Numbers - R : All rational and irrational numbers

• X-Intercepts - The point(s) where a graph intersects the x-axis (when y=0)

• Y-Intercepts - The point(s) where a graph intersects the y-axis (when x=0)

• Commutative Property of Addition - a + b = b + a

• Commutative Property of Multiplication - ab = ba

• Associative Property of Addition - a + (b + c) = (a + b) + c

• Associative Property of Multiplication - a(bc) = (ab)c

• Distributive Property - a(b + c) = ab + ac

• Zero Property of Multiplication - (a)(0) = 0

• Inadmissible - Values that cannot be used

• Absolute Value - The distance from zero on a number line (always positive), shown by | |

• Opposites - Two values with the same absolute value but different signs

• Reciprocals - Two numbers whose product is 1. One value is equivalent to 1 over the the other value

• Translation - A transformation that causes all points on a graph to move the same amount in the same direction, which produces a congruent image in the same orientation

• Dilation - Stretches or shrinks that change the shape of the original graph by moving values closer or farther from zero

• Reflection - Transformations that move points to the opposite side of zero and may change the orientation

• Invariant Point - A point that is not affected by a transformation

• Asymptote - A line that a graph approaches but never crosses

• Axis of Symmetry - The line on a graph where both sides are symmetrical

• Inverse of a Function - Another relation that still connects the same two variables but reverses the mappings of the original function.

• One-to-One Function - If different inputs always produce different outputs, and inputs can only be the same if outputs are the same

• Increasing Function - A graph where x gets bigger as y gets bigger, or as x gets smaller, y gets smaller

• Decreasing Function - A graph where x gets bigger as y gets smaller, or as x gets smaller, y gets bigger

• Argument of a Function - The value that is inputted

• Odd Function - A function where every point in the domain has a mirror point with opposite x and y-values. f(-x) = -f(x)

• Even Function - A function with every point in the domain having a mirror point with an opposite x-value. f(x) = f(-x)

• Index - The root which is being taken

• Radicand - The number of which the root is taken

• Base - The number which is being multiplied by itself

• Exponent - The number of times that a number is multiplied by itself

• Linear Growth - A value that increases by a constant difference

• Linear Decay - A value that decreases by a constant difference

• Exponential Growth - A long period of relatively little change followed by explosive, exponential growth

• Exponential Decay - Dramatic, exponential decreases followed by a long period of relatively little change

• Half-Life - The amount of time it takes to end up with half of the starting amount

• Doubling Time - The amount of time it takes to end up with double the starting amount

• Hypotenuse - The long side of a right triangle

• Legs - The two sides adjacent to the hypotenuse and right angle

• Ratio - The relationship in quantity between two values

• Angle - An amount of rotation around a point called the vertex

• Initial Arm - Where the rotation starts. On the positive x-axis for standard position

• Terminal Arm - Where the rotation ends

• Coterminal Angles - Angles with measures that differ by a multiple of 360∘ or 2π. In standard position, coterminal angles have the same terminal arm

• Reference Angle - The acute angle between the terminal arm and the x-axis

• Special Angle - Angles in trigonometry that can be expressed as an exact value of its ratio

• Sine Ratio - y/r

• Cosine Ratio - x/r

• Tangent Ratio - y/x

• Cosecant Ratio - r/y

• Secant Ratio - r/x

• Cotangent Ratio - x/y

• Angle in Standard Position - An angle that has its vertex at the origin and its initial arm on the positive x-axis

• Identity - Equations that are true for every value of the variable(s) for which they are defined

• Periodic Function - A function whose values repeat at set intervals of the independent variable values

• Period of a Function - The horizontal length of one cycle of a periodic function

• Amplitude - Half the vertical distance from the max to the min of a periodic function

• Sinusoidal Function - A function that repeats in a smooth wave pattern

• Phase Shift - A horizontal shift of a periodic function

• Principal Amount - The amount of money initially invested

• Interest - The amount of money earned by an investment/paid on a loan

• Accrued Amount - The total of the original principal plus interest

• Present Value - The amount of money initially invested or borrowed

• Future Value - The total of the original principal amount plus interest

• Simple Interest - Interest earned/paid on the original principle only

• Compound Interest - Interest earned/paid on both the original principle and any previously earned interest

• Compounding Period - The amount of time that passes before interest is compounded

Unit 1: Algebra Review and Extension

Factoring

• Always try to common factor first

• A number with factors of one and itself are prime

• Sum of Two Cubes - a³+b³= (a + b)(a² - ab + b²)

• Difference of Two Squares - a²-b² = (a + b)(a - b)

• Difference of Two Cubes - a³- b³ = (a - b)(a² + ab + b²)

• Perfect Square Trinomial - (a + b)² = a² + 2ab + b²

• Decomposition - ax² + bx + c = ax² + mx + bx + c

  • Where mx + nx = bx, and (mx)(nx) = (ax²)(c)

Radicals

• nth root - a number c is the nth root of a if cⁿ = a, or ⁿ√a = c

• Even roots - the radicand cannot be negative, or else there are no real numbers

  • When taking the even root of a positive radicand, there is a negative and positive root

  • If there is no ± sign, it is assumed to be the principal root (non-negative roots only)

• Odd roots - the root will have the same sign as the radicand, and only one possible root

• Properties of radicals

  even n, n√an = |n|	(n√a)(n√b) = n√ab	n√am = (n√a)m
  odd n, n√an = n	n√(a/b) = (n√a)/(n√b), b≠0	*n = the nth root being taken

• Reducing radicals

  • Break a single radical into two radicals, one of which can be evaluated

  • The radicand should not contain any factors that have degrees greater than or equal to the index

• Adding/subtracting

  • Combine like terms

• Multiplying/dividing

  • If they have the same indices, multiple the radicands and combine

• Rationalizing the denominator

  • Monomial denominator - multiply the top and bottom by just the square root

  • Binomial denominator - multiply the top and bottom by the denominator’s conjugate

Equivalent Expressions and Formulas

• and - both must be true; or - at least one must be true

• To check if two are equal, simplify both and check if they are exactly the same

• Multiplying or dividing

  • For division, multiply by the reciprocal

  • Factor the numerator and denominator

  • Cancel common factors

    • Consider opposites - a negative 1 can be factored out to make them equal

  • Multiply numerators and denominators

  • State all restrictions based on the original expression using AND

    • For division, consider the numerator and denominator of the second factor

• Adding or subtracting

  • Factor each numerator and denominator

  • Reduce each individually

  • Compare the denominators - multiply the fractions by terms that are not common

  • Add or subtract

    • For subtracting, be sure to apply the negative to the whole numerator

  • Reduce

  • State restrictions

Absolute Value

• |m| is the absolute value and a grouping symbol, so they should be treated like brackets

• |m| = m if m is non-negative

• |m| = -m if m is negative

• |m| = -m if m is non-negative, there are no solutions

• Properties of absolute values

  |a| = |-a|	|ab| = (|a|)(|b|)
  |a/b| = |a|/|b|, b ≠ 0	|a + b| ≤ |a| + |b|

• Solving linear absolute values

  • Isolate the absolute value

  • Break it into two equations

  • Solve each equation

Unit 2: Fundamentals for Working with Functions

Function Review

• x and f(x) or y are used to represent relations with no context

• In a function, one input will always produce one output

  • If independent values repeat, it is not a function

  • If more than one arrow goes from one independent value, it is not a function

  • On a graph, if a vertical line drawn at any point can intercept more than one point, it is not a function

    • Explain the points that can be intercepted, what this says about x-coordinates, and conclude if it is not a function

  • In an equation, isolate the dependent variable - if the independent variable can produce more than one result (ex. even roots, absolute value), then it is not a function

• Function notation: f(x) is “f of x” or “f at x”

One-to-One Functions

• If a ≠ b, f(a) ≠ f(b)

• If f(a) = f(b), then a = b

• Neither the input or output can repeat

  • Must be a function that does not produce repeating dependent variables

  • On a graph, use a horizontal line test if it passes the vertical line test

• Solving one-to-one functions algebraically, if x-values aren’t known

  • “If f(a) = f(b)”

  • Use the formula for f(x) to substitute in (a) and (b)

  • Isolate a, and show each step

  • If a = b, this is one-to-one, and if a ≠ b, it is not one-to one

  • Conclude: “When f(a) = f(b), the conclusion is that [a = b] or [a ≠ b]. Therefore, f(x) is [not] one-to-one.”

• Solving one-to-one functions algebraically, if two known inequal x-values produce different y’s

  • “If f(a) = f(b):”

  • Input each of the x-values

  • Isolate for each individually

  • Conclude: “a ≠ b, but f(a) = f(b). Therefore, f(x) is not one-to-one.”

Parent Functions

Intervals of Increase and Decrease

• Increasing function: when x₁ > x₂, f(x₁) > f(x₂)

• Increasing function: when x₁ > x₂, f(x₁) < f(x₂)

• Input values from the domain

Even and Odd Functions

• Even functions are symmetrical along the y-axis

• Odd functions are the same when rotated 180∘ around the origin

• To solve algebraically:

  • Simplify f(x) and f(-x)

  • Compare f(-x) and f(x). If they are equal, it is an even function

  • Conclude that “f(x) =/≠ f(-x). Therefore, this is [not] even.”

  • Find -f(x), simplify it, and compare it to f(x). If they are equal, it is odd.

  • Conclude that “f(x) =/≠ -f(x). Therefore, this is [not] odd.”

  • If it is neither odd nor even, it is neither

Inverses of Functions

• When the mappings (independent and dependent) are reversed

  • The range of a function becomes the domain of the inverse, and vice versa

  • Inverse of a function is not always an inverse function (occurs in one-to-one functions)

• Graphs are flipped across the line y = x

  • Pick key anchor points and swap their coordinates

• Solving given context:

  • Write the function’s formula

  • Isolate for the independent variable, making it now the dependent

  • Considering the equation and context, decide the inverse domain and range

    • The values of the domain of the function will have the same values in the inverse, but instead as the range, and vice versa

    • In some cases, values may only be non-negative, so input 0 into the equation and solve for the restriction

  • Input any values to solve

• Solving without context:

  • Swap the x’s and y’s

  • Isolate for y

• Algebraic test for inverses

  • f(x) and g(x) are inverses iff f(g(x)) = x and g(f(x)) = x

  • Input one equation into the other and simplify

  • If both equal x, conclude that they are inverses

Unit 3: Transformations of Functions

Translations

• y = f(x - h) + v → (x + h, y + v)

• All x’s are replaced with x - h, and a horizontal shift of h units occurs

  • h will appear as its opposite in the equation, so shift it the opposite direction

• v is added to all y-coordinates, and a vertical shift of v units occurs

• Alter the points from the parent formula, or plot the point of reference (h, v) and plot the parent function as if it were the origin

Reflections and Dilations

• y = af(kx) → [(1/k)x, ay]

  • Graphs are horizontally stretched or shrunk by |1/k|, and may be reflected

  • Graphs are vertically stretched or shrunk by |a|, and may be reflected

• Graphs are stretched by a factor > 1

• Graphs are shrunk by a factor < 1

Combinations of Transformations

• y = af[k(x - h)] + v → [(1/k)x + h, ay + v]

• y = af[kx - kh)] + v → [(1/k)(x + kh), ay + v]

• When stating horizontal transformations:

  • If the argument is factored, state dilations first, then shifts

  • If the argument is not factored, state shifts first, then dilations

• Vertical transformations affect the function as a whole

• Horizontal transformations use the opposite operations

• Creating accurate sketches:

  • Determine the point of reference

  • Factor the argument if necessary

  • Apply reflections and dilations only to the parent points

• When giving equivalent formulas, consider fractions, factoring the argument (or not), separating and evaluating radicals, etc.

Unit 4: Exponential Functions and Their Inverses

Exponent Rules

• If the base is a constant, work in radical form

• If the base is a variable, work in exponential form

• The exponent only applies to the factor that they directly follow

Introduction to Logarithms

• log_ba is “logarithm base b of a,” where b is the base and a is the argument

• y = log_bx iff b^y = x, b > 0, and b ≠ 1

  • y = log_bx is in logarithmic form

  • b^y = x is in exponential form

• log_bx is the exponent that must be applied to b to get a value of x

• If the base is not written, the convention is that it is 10 (common logarithms)

• If the base is e, it is a natural logarithm written as ln

Logarithm Rules

Parent Exponential Functions

• Formulas that match f(x) = b^x, b > 0, and b ≠ 1

• Consecutive values of the finite differences change by a constant factor (excluding 1, 0, -1). This means that the ratios of every pair of consecutive finite differences are the same

• The formula for the function has the independent variable in the exponent of a power with a constant base

• Long periods of almost no change in the function values, followed by faster and faster increase or decrease

Solving Exponential Equations

• Strategy 1: A single power on each side with the same base

  • The exponents must be equal

• Strategy 2: Convert an exponential equation into a logarithmic equation

• Strategy 3: Take logarithms using the same base of both sides, and then use the power rule for logarithms

Exponential Growth and Decay

• To change by a constant factor

• Exponential growth is little change followed by explosive growth

• Exponential decay is characterized by a dramatic decrease followed by a long period of relatively little change

• In a word problem, the first factor will usually be the base, the second factor will be the base multiplied by the constant factor to the first exponent, and then the exponent will continue to grow

• To solve:

  • Establish that it is exponential

  • Use A_c = A_IF^t/p

    • A_c = Current Amount

    • A_I = Initial amount (t=0)

    • F = Growth/decay factor

      • For half-life questions: F = ½

      • For doubling questions: F = 2

      • For losing an amount by n percent: F = 1 - (n)/100

      • For gaining an amount by n percent: F = 1 + (n/100)

      • For compounded interest of n percent: F = 1 + [n/(t/p)]

      • If only given 2 values, divide the second by the first to find F

    • t = elapsed time

    • p = length of one unit

Logarithmic Functions

• Any equation that follows y = log_bx where b > 0 and b ≠ 1

• First list the ordered pairs for the parent exponential function

• Then switch the domain and range to get the inverse

• Graph following graphing standards

Unit 5: Trigonometry Review and Extensions

Review of Triangle Trigonometry

• A triangle has 3 side lengths and 3 angles

• To solve, at least one side length and two other measurements must be known

  Sine sin x = opp/hyp	Cosine cos x = adj/hyp	Tangent tan x = opp/adj
  Cosecant csc x = hyp/opp	Secant sec x = hyp/adj	Cotangent cot x = adj/opp
  Angle Sum of a Triangle a1 + a2 + a3 = 180∘	Pythagorean Theorem a² + b² = c²	Sine Law sin A/a = sin B/b = sin C/c
  Cosine Law c² = a² + b² - 2ab(cos C)

• To solve, write down the applicable formula, and substitute appropriate variables

• The tool must work with the proper triangle

  • Ratios and Pythagorean Theorem can only be used on right triangles

• If there are multiple triangles, identify which one you are working in

2D and 3D Problem Solving

• Angle of depression - angle that goes down from a parallel line

• Angle of elevation - angle that goes up from a parallel line

• Cardinal directions are North, East, South, West

• Bearing is the angle in one direction that moves towards one of the adjacent directions

• True bearing - clockwise direction from true north

• To solve problems

  • Draw and label neat diagrams with vertices and known measurements

  • Identify the RTF

  • Choose the proper tools and establish which triangle is being worked on

  • Use precise values

  • Consider the ambiguous case

Units of Measure, Coterminal Angles, Reference Angles

• degrees and radians measure angles

• One complete rotation is 360∘ or 2π

• A positive angle measure is a counter-clockwise rotation

• Conversions

  • 1∘ = π/180

  • 1 = (180/π)∘

  • Multiply the fraction by the number of degrees or radians that are given

• To find coterminal angles, add one rotation in units

• To find the reference angle, use the quadrantal angles to determine the angle within a quadrant, and then isolate for the reference angle

Trigonometry of Angles of Rotation

• The trigonometric ratios are as followed, where (x, y) is any point on θ’s terminal arm, and r is the (positive) distance from the point to the origin

  sin θ = y/r	cos θ = x/r	tan θ = y/x
  csc θ = r/y	sec θ = r/x	cot θ = x/y

• Distance equation:

  • r = √(x² + y²)

• To solve for ratios, take the given values, solve for the unknown value, and put the result into the ratio definitions

Related Angles

• Related angles - angles that are in standard position with terminal arms that have the same reference angles

• Principle angles - the first non-negative angle in standard position

• Quadrantal Angles - angles in standard position that stop on an axis

• Directed arc - an arc with an arrow indicating direction and amount of rotation

• Let θ represent the reference angle

  Quadrant	Coordinates	Sine	Cosine	Tangent	Formula
  1	(x, y)	Positive	Positive	Positive	θ
  2	(-x, y)	Positive	Negative	Negative	180∘ - θ
  3	(-x, -y)	Negative	Negative	Positive	180∘ + θ
  4	(x, -y)	Negative	Positive	Negative	360∘ - θ

The Ambiguous Case

• Sine ratios are always between 1 and -1

• Related angles with the same sine ratio will stop in the first and second quadrant

• Solving triangles means to find all missing measures

• The ambiguous case:

  • Given two side lengths and the angle opposite one of the sides

  • The given and is acute

  • The side opposite the given angle is shorter than the other given side

  • To solve:

    • Use the sine law to find the reference angle, and then the related angle that falls in quadrant 2

    • Draw a diagram of the two possible triangles

    • To find other angle measures, use the sum of angles of a triangle theorem

    • To find other sides, use sin law

    • There will be two possible answers, as there is an obtuse and acute angle

Unit 6: Trigonometric Functions and Equations

Functions

• For sine and cosine, always plot the MMM’s and IBs

  • There will be an MMM every quarter of a period

  • An IB will be 2/3 away from the max or min, and 1/3 away from a mid

• For tangent, plot asymptotes

Solving Trigonometric Equations

• The unknown variable that must be solved is the argument

• For simple arguments:

  • First solve for the ratio - isolate the function on one side

  • Based on the ratio, identify which quadrant the terminal arm(s) are in, and if it is a positive, negative, undefined, or 0 value

  • Draw a diagram

  • Identify the reference angle, using special triangles or arc-functions

  • Determine the principal angles

  • Find coterminal angles to fulfill the restrictions

• For complex arguments:

  • Solve for the ratio

  • Replace the entire argument with just one variable

  • Based on the ratio, identify which quadrant the terminal arm(s) are in, and if it is a positive, negative, undefined, or 0 value

  • Draw a diagram

  • Identify the reference angle, using special triangles or arc-functions

  • Determine the principle angles

  • Input the principle angles into the entire original argument and solve

  • Multiply the parent period by the horizontal stretch/shrink factor

  • Add and subtract this number to the transformed principle angles to find all applicable angles

Modelling with Sinusoidal Functions

• To find equations:

  • Using the values above, find the variables

  • Before finding the phase shift variables, determine whether reflections will apply, and thus, if |a| and |k| will be positive or negative values

  • Find the phase shifts

  • Use the variable values to create a transformed parent equation

Tangent Functions

• To transform asymptotes:

  • Find the location of one asymptote using: x = (1/k)(90∘) + h

  • Find the period using: period = 180∘/|k|

  • The domain cannot be the location of that one asymptote, ± the period

Unit 7: Trigonometric Identities

Properties of Addition and Multiplication

Proofs of Identities

• Separate the equation into its left and right side

• Make substitutions and simplify, showing all steps and justifying

  • Try factoring and manipulating algebraically when stuck

• Once the two sides are equal, state:

  • “LS = RS. Therefore, this is an identity!!!”

  Reciprocal Identity csc x = 1/sin x	Reciprocal Identity sec x = 1/cos x	Reciprocal Identity cot x = 1/tan x
  Quotient Identity tan x = sin x/cos x	Quotient Identity cot x = cos x/sin x	Pythagorean Identity sin²x + cos²x = 1

Addition Identities and Special Triangles

• Angles can be written as a sum of other angle measures

• Using addition identities, 15∘ (60-45 or 45-30) and 75∘ (45 + 30) become special angles

  • Solve for the ratio, and apply negatives or positives as the quadrant requires

Unit 8: Discrete Functions and Financial Applications

Arithmetic and Geometric Sequences

• A sequence is a set of values in a specific order, usually connected by some sort of pattern or rule

• A term is a single value, and the term number indicates the position of a term in a sequence

• t_n is used to refer to the nth term of a sequence

• The general term is the formula that can be used to calculate the value of any term in the sequence, where n is the independent variable

• An explicit formula for the general term is one that can be used to calculate the values of terms in a sequence based only on n

• Arithmetic Sequence - A sequence where the difference between consecutive terms (the common difference) is constant

  • a is the value of the first term

  • d is the common difference (t_n - t_n-1)

    Term #	1	2	3	4	…
    Term Value	a	a + 1d	a + 2d	a + 3d	…

  • t_n = a + (n - 1)d, n∈N

• Geometric Sequence - A sequence where the ratio of consecutive terms (the common ratio) is constant

  • a is the value of the first term

  • r is the common ratio (t_n/t_n-1)

    Term #	1	2	3	4	…
    Term Value	a	ar1	ar²	ar³	…

  • t_n = ar^n-1, n∈N

• To solve, first prove that it is arithmetic/geometric using the common difference/ratio. Then, use the equation

Arithmetic and Geometric Series

• A series is the sum of terms in a sequence

• S_n means the sum of the first n terms

• The arithmetic series of n terms is: S_n = n/2 [a + t_n], or S_n = n/2 [2a + (n-1)d]

• The geometric series of n terms is: S_n = a(rⁿ - 1)/ (r - 1)

• Use the variables given and input them

• Sometimes, n will have to be solved through a sequence first

Interest Calculations

• Simple interest is arithmetic, and compounded interest is geometric

• Simple interest

  • A = P + Prt

  • I = Prt

    • I is the amount of interest, P is the principal, r is the annual interest rate, t is the time elapsed in years, and A is the accrued amount

• Compound interest

  • A = P(1 + i)ⁿ

  • PV = (FV)/(1 + i)ⁿ

  • I = A - P

• Variables:

  • I - amount of interest

  • P/PV - principal

  • r - annual interest rate

  • t - time elapsed in years

  • A/FV - accrued amount

  • i - interest rate per period (annual rate/periods per year)

  • n - time elapsed in number of periods

Annuities

• An annuity is a sum of payments

• Accrued annuity - payments are made over time to collect a larger fund

• Present value annuity - When a sum of money is put in and payments come out over time

• Simple ordinary annuity - the interval between payments is the same as compounding periods, and payments are made at the end of each period

• Timelines

  • Make payments at the end of each period

  • The last payment should not have any factors besides the initial payment

  • The first payment should be n-1

• Simple, Ordinary Annuity Formulas

  • A = R[(1 + i)ⁿ -1]/i

  • A - accrued amount

  • R - regular payment

  • i - interest rate per compounding period

  • n - number of compounding periods