MCR3U · Exam review

Functions Exam Review

A practice exam with questions from all 4 units, then the key facts from every lesson, unit by unit. For a deeper look at one unit, open its test review.

Unit test reviews

Practice exam

32 questions, the same number from each unit. Each one shows the lesson it comes from.

Unit 1: Relations vs. Functions
1
Which set of ordered pairs is NOT a function?
Unit 1: Domain and Range
2
A rock dropped from a cliff has height $h (t) = 80 - 5 t^{2}$ metres. What is the largest value in the domain of this model (the time it lands), in seconds?
s
Unit 1: Transformations of Functions
3
State the range of $g (x) = - 3 x + 2 + 7$ as an inequality.
Unit 1: Forms of a Quadratic Relation
4
For $y = 5 (x + 1) (x - 4)$ , what are the $x$ -intercepts?
Unit 1: Completing the Square
5
Which is the completed-square form of $y = x^{2} + 5 x + 1$ ?
Unit 1: Quadratic Applications
6
A ball's height is $h (t) = - 4.9 t^{2} + 24.5 t + 2$ . To the nearest 0.1 s, at what time is the ball at its maximum height?
s
Unit 1: Rational Expressions
7
Simplify $\frac{x ^{2} - 16}{x + 4}$ and state the restriction.
Unit 1: Equivalent Algebraic Expressions
8
Which expression is NOT equivalent to $2 x^{2} - 5 x - 12$ ?
Unit 2: Exponent Laws
9
Evaluate $1 6^{3/4}$ .
Unit 2: Rational Exponents and Radicals
10
Evaluate $8 1^{3/4}$ without a calculator.
Unit 2: Graphs of Exponential Functions
11
The graphs of $y = 2^{x}$ and $y = x^{2}$ meet at $x = 2$ and at one larger whole number. What is that larger value of $x$ ?
Unit 2: Linear, Quadratic or Exponential?
12
Each data set was measured at equal time intervals. Which is best modelled by an exponential function?
Unit 2: Transformations of Exponential Functions
13
Reflecting y = 2ˣ in the y-axis produces a graph identical to which function?
Unit 2: Finding an Exponential Equation
14
An exponential function passes through $(0, 3)$ and $(1, 6)$ . Write its equation in the form $y = a (b)^{x}$ .
Unit 2: Solving Exponential Equations
15
Solve $2 7^{2 x} = 9^{x + 4}$ .
Unit 2: Exponential Growth and Decay Applications
16
A bacteria culture of 500 cells doubles every 20 minutes. After how many minutes will there be 16000 cells?
min
Unit 3: Sequences as Discrete Functions
17
For $t_{n} = 2 n + 1$ , what is $t_{5}$ ?
Unit 3: Arithmetic Sequences
18
An arithmetic sequence has $a = 7$ and $d = 4$ . Find $t_{12}$ .
Unit 3: Geometric Sequences
19
Using $r = 2$ and $t_{3} = 20$ from the previous question, find $a$ .
Unit 3: Arithmetic Series
20
Find the sum of the first 40 positive even numbers ( $2 + 4 + 6 + \dots + 80$ ).
Unit 3: Geometric Series
21
Every term of a sequence is $12$ (constant, so $r = 1$ ). Find $S_{9}$ .
Unit 3: Pascal's Triangle and the Binomial Theorem
22
Expand $(x + 2)^{4}$ (from the worked example) and evaluate it at $x = 1$ to check. What value do you get?
Unit 3: Simple and Compound Interest
23
For 8%/year over 3 years, match each compounding frequency to its values of i and n.
1. A. $i = \frac{0.08}{12}, n = 36$
2. B. $i = 0.02, n = 12$
3. C. $i = 0.04, n = 6$
4. D. $i = 0.08, n = 3$
Annually
Semi-annually
Quarterly
Monthly
Unit 3: Simple Annuities
24
Match each question to the quantity you should calculate.
1. A. $R$ from the $F V$ formula
2. B. $R$ from the $P V$ formula
3. C.Future value $F V$
4. D.Present value $P V$
How much will my monthly deposits grow to?
How big a loan can these payments repay?
What deposit reaches my savings goal?
What payment pays off this loan?
Unit 4: Right Triangle Trigonometry
25
A right triangle has hypotenuse $12$ and an angle of $25°$ . Find the length of the side opposite that angle (round to 2 decimals).
Unit 4: Special Triangles and Exact Values
26
Match each expression to its exact value.
1. A. $\frac{2}{2}$
2. B. $\frac{1}{2}$
3. C. $3$
4. D. $0$
5. E. $2$
$sin 30°$
$cos 45°$
$tan 60°$
$csc 30°$
$cos 90°$
Unit 4: Angles from 0° to 360° and the CAST Rule
27
Put the steps for solving $tan θ = - 1.2$ on $0° \leq θ < 360°$ in order, taking the quadrants counterclockwise.
1. 1Check both answers on the calculator
2. 2QIV: $θ = 360° - 50.2° = 309.8°$
3. 3QII: $θ = 180° - 50.2° = 129.8°$
4. 4Find the reference angle $tan^{- 1} (1.2) \approx 50.2°$
5. 5Use CAST to place $50.2°$ : tangent is negative in QII and QIV
Unit 4: The Ambiguous Case of the Sine Law (SSA)
28
In $△ A BC$ , $A = 40°, a = 12, b = 8$ . How many triangles are possible?
Unit 4: The Cosine Law
29
Put the steps for solving an SSS triangle in the safest order.
1. 1Check that larger angles sit opposite longer sides
2. 2Find the third angle from the $180°$ angle sum
3. 3Use the sine law for a second angle
4. 4Identify the longest side
5. 5Use the cosine law for the angle opposite it
Unit 4: Periodic Functions: Period, Amplitude, and Axis
30
A graph has maximum $50$ and minimum $10$ . Find the amplitude.
Unit 4: Graphs of y = sin x and y = cos x — Transformations
31
Find the period of $y = sin (4 x)$ .
°
Unit 4: Modelling with Sinusoidal Functions
32
Put the steps of this lesson's translation checklist for modelling a situation with a sinusoid in order.
1. 1Check the model at a known point
2. 2Choose $sin$ , $cos$ or $- cos$ and $d$ from the starting behaviour
3. 3Find the period and compute $k = \frac{360°}{period}$
4. 4Identify the maximum and minimum values
5. 5Compute the axis $c$ and amplitude $∣ a ∣$

Key facts by unit

Unit 1: Characteristics of Functions

Relations vs. Functions

A relation is a function iff every input has exactly one output (no repeated $x$ with different $y$ ).
Vertical line test: if a vertical line crosses the graph more than once, it is not a function.
$f (x)$ names the output of $f$ at input $x$ ; it is not $f \times x$ .
To evaluate $f (a)$ , replace every $x$ in the rule with $a$ , then simplify.
Independent variable = input = $x$ (domain); dependent variable = output = $f (x)$ or $y$ (range).
To solve $f (x) = k$ , set the rule equal to $k$ and solve for $x$ ; a many-to-one function can give several solutions.

Domain and Range

Domain = every valid input ( $x$ ); Range = every possible output ( $y$ ).
Radical restriction: the radicand of an even root must be $\geq 0$ .
Denominator restriction: a denominator can never equal $0$ .
Interval notation: square bracket $[]$ = included endpoint, round bracket $()$ = excluded endpoint; $\pm \infty$ always gets a round bracket.
On a graph, domain is read along the $x$ -axis and range along the $y$ -axis.
For $y = a (x - h)^{2} + k$ , the range is $y \geq k$ when $a > 0$ and $y \leq k$ when $a < 0$ .

Parent Functions

$y = x$ : domain and range are all real numbers; it's the only parent function that's a straight line.
$y = x^{2}$ : domain all reals, range $y \geq 0$ ; vertex (turning point) at the origin.
$y = x$ : domain $x \geq 0$ , range $y \geq 0$ ; starts at the origin.
$y = \frac{1}{x}$ : domain and range both exclude $0$ ; has asymptotes along both axes.
$y = ∣ x ∣$ : domain all reals, range $y \geq 0$ ; sharp corner (not smooth) at the origin.
$y = x$ is the right half of $y = x^{2}$ reflected in $y = x$ : key points $(0, 0), (1, 1), (4, 2), (9, 3)$ .

Transformations of Functions

$y = a f (k (x - d)) + c$ : $a$ = vertical stretch/reflection, $k$ = horizontal stretch/reflection, $d$ = horizontal shift, $c$ = vertical shift.
Mapping rule: $(x, y) \to (\frac{x}{k} + d, a y + c)$ .
Negative $a$ reflects in the $x$ -axis; negative $k$ reflects in the $y$ -axis.
Horizontal stretch/compression factor is $\frac{1}{k}$ (the reciprocal of $k$ ), not $k$ itself.
Apply stretches and reflections before translations — that's exactly the order built into the mapping rule.
Factor $k$ out of the bracket before reading $d$ : $f (2 x - 6) = f (2 (x - 3))$ gives $d = 3$ .

Inverse Functions

To find $f^{- 1} (x)$ algebraically: replace $f (x)$ with $y$ , swap $x$ and $y$ , then solve for $y$ .
$f$ and $f^{- 1}$ are reflections of each other in the line $y = x$ ; if $(a, b)$ is on $f$ , then $(b, a)$ is on $f^{- 1}$ .
$f^{- 1} (x)$ is notation for the inverse function, not $\frac{1}{f ( x )}$ .
The inverse of $f$ is a function only if $f$ passes the horizontal line test (i.e., $f$ is one-to-one).
Restricting the domain of a many-to-one function (like $x^{2}$ ) can make its inverse a function.
Domain of $f^{- 1}$ = range of $f$ , and range of $f^{- 1}$ = domain of $f$ ; verify with $f (f^{- 1} (x)) = x$ .

Forms of a Quadratic Relation

Standard form $y = a x^{2} + b x + c$ : $c$ is the $y$ -intercept, sign of $a$ gives opening direction.
Factored form $y = a (x - r_{1}) (x - r_{2})$ : $r_{1}, r_{2}$ are the $x$ -intercepts.
Vertex form $y = a (x - h)^{2} + k$ : $(h, k)$ is the vertex (max/min point).
Axis of symmetry from factored form: $x = \frac{r _{1} + r _{2}}{2}$ ; from vertex form: $x = h$ ; from standard form: $x = \frac{- b}{2 a}$ .
All three forms describe the same parabola — converting between them never changes the graph, only how it's written.

Completing the Square

To complete the square on $x^{2} + b x$ : add and subtract $(\frac{b}{2})^{2}$ .
When $a \neq = 1$ , factor $a$ out of the $x^{2}$ and $x$ terms first, before completing the square.
After completing the square, the constant subtracted outside the bracket must be multiplied by $a$ .
Vertex form result: $y = a (x - h)^{2} + k$ , vertex $(h, k)$ .
If $a > 0$ , $k$ is a minimum value; if $a < 0$ , $k$ is a maximum value.

Solving Quadratic Equations

Zero product property: if $A \cdot B = 0$ then $A = 0$ or $B = 0$ — only valid when one side of the equation is exactly $0$ .
Quadratic formula: $x = \frac{- b \pm b ^{2} - 4 a c}{2 a}$ , valid for any $a x^{2} + b x + c = 0$ .
Discriminant: $D = b^{2} - 4 a c$ .
$D > 0$ : two real roots; $D = 0$ : one repeated real root; $D < 0$ : no real roots.
Before factoring or using the formula, always set the equation equal to $0$ first.

Quadratic Applications

Projectile height model: $h (t) = - 4.9 t^{2} + v_{0} t + h_{0}$ (metres, seconds).
Revenue model: $Revenue = price \times quantity$ .
The vertex of a quadratic model gives the maximum (if $a < 0$ ) or minimum (if $a > 0$ ) value of the situation.
The $t$ - or $x$ -coordinate of the vertex tells you when or at what input the max/min happens; the $y$ -coordinate tells you the max/min value itself.
Always interpret the vertex back in context — include correct units in the final answer.

Radicals

Simplify an entire radical by factoring out the largest perfect square: $72 = 62$ .
Only like radicals combine: $a n + b n = (a + b) n$ .
$a \cdot b = ab$ ; multiply coefficients and radicands separately.
Rationalize a single-term radical denominator by multiplying top and bottom by that radical.
Rationalize a binomial radical denominator by multiplying top and bottom by its conjugate: $(A + B) (A - B) = A^{2} - B^{2}$ .

Rational Expressions

Always factor completely before cancelling, multiplying, dividing, or finding a common denominator.
Restrictions come from every denominator in the original (un-cancelled) expression.
Dividing rational expressions: multiply by the reciprocal of the second fraction.
Adding/subtracting rational expressions: rewrite over a common denominator, then combine numerators only.
A cancelled factor still creates a restriction — cancelling changes the simplified form, not the domain.

Equivalent Algebraic Expressions

Two expressions are equivalent if and only if they produce the same output for every value of the variable.
To prove equivalence, fully expand and simplify both expressions, then compare term by term.
A single matching test value is not proof of equivalence — it can only disprove it (if they disagree there).
$(A + B)^{2} = A^{2} + 2 A B + B^{2}$ — the middle term is easy to forget.
$(A - B) (A + B) = A^{2} - B^{2}$ (difference of squares) — this is different from $(A - B)^{2}$ .

Unit 2: Exponential Functions

Exponent Laws

Product law: $a^{m} \cdot a^{n} = a^{m + n}$ (same base — add exponents).
Quotient law: $\frac{a ^{m}}{a ^{n}} = a^{m - n}$ (same base — subtract exponents).
Power of a power: $(a^{m})^{n} = a^{mn}$ (multiply exponents).
Zero exponent: $a^{0} = 1$ for any $a \neq = 0$ . Negative exponent: $a^{- n} = \frac{1}{a ^{n}}$ .
Rational exponent: $a^{m / n} = (n a)^{m}$ — denominator is the root, numerator is the power.
Power of a product or quotient: $(ab)^{n} = a^{n} b^{n}$ and $(\frac{a}{b})^{n} = \frac{a ^{n}}{b ^{n}}$ — but $(a + b)^{n} \neq = a^{n} + b^{n}$ .

Rational Exponents and Radicals

$a^{1/ n} = n a$ , the principal (non-negative) root when $n$ is even.
$a^{m / n} = (n a)^{m} = n a^{m}$ : the denominator is the root, the numerator is the power.
$a^{- m / n} = \frac{1}{a ^{m / n}}$ and $(\frac{a}{b})^{- m / n} = (\frac{b}{a})^{m / n}$ : flip first, and the value stays positive.
An even root of a negative number is not real; an odd root of a negative is negative: $(- 27)^{1/3} = - 3$ .
By hand, take the root before the power: $2 7^{4/3} = 3^{4} = 81$ .
The exponent laws still apply; add, subtract or multiply the fractional exponents (for $x > 0$ ).

Graphs of Exponential Functions

$y = b^{x}$ , $b > 0$ , $b \neq = 1$ : passes through $(0, 1)$ , horizontal asymptote $y = 0$ .
$b > 1$ : exponential growth (increasing). $0 < b < 1$ : exponential decay (decreasing).
Domain of $y = b^{x}$ : all real numbers. Range: $y > 0$ .
An exponential growth function ( $b > 1$ ) eventually exceeds any polynomial function, no matter the polynomial's degree.
$b = 1$ is excluded since $1^{x} = 1$ is constant, not exponential.
In a table with equal $x$ -steps: constant first differences mean linear, constant ratios mean exponential (the ratio is $b$ when $x$ steps by 1).

Linear, Quadratic or Exponential?

With equal $x$ -steps: constant first differences mean linear, constant second differences mean quadratic, constant ratios mean exponential.
An exponential's differences are exponential too: $a b^{x + 1} - a b^{x} = (b - 1) a b^{x},$ so every column of differences has the same ratio $b$ and is never constant.
For $y = a x^{2} + b x + c$ with $x$ -steps of 1, the second difference is $2 a$ . With a step of $h$ : linear differences are $mh$ , quadratic second differences $2 a h^{2}$ , exponential ratios $b^{h}$ .
Uneven $x$ -steps: compare $\frac{Δ y}{Δ x}$ for a line, and take the $h$ -th root of a ratio across a step of $h$ for an exponential.
Real data are exponential if the ratios are roughly constant with no trend; ratios drifting steadily toward 1 point to a polynomial.
An exponential with $b > 1$ eventually exceeds every linear and quadratic function, because its step-to-step factor stays $b$ while theirs shrinks toward 1.

Transformations of Exponential Functions

$y = a \cdot b^{k (x - d)} + c$ : same parameter roles as any transformed function.
The horizontal asymptote is always $y = c$ — read it directly off the equation.
Find the $y$ -intercept by substituting $x = 0$ and simplifying — there is no general shortcut.
$a < 0$ reflects in the $x$ -axis; $k < 0$ reflects in the $y$ -axis.
$2^{- x} = (1/2)^{x}$ — reflecting a growth curve in the $y$ -axis produces a decay-looking curve with a reciprocal base.
Mapping rule: $(x, y) \to (\frac{x}{k} + d, a y + c)$ . Range: $y > c$ if $a > 0$ , $y < c$ if $a < 0$ .

Finding an Exponential Equation

From the $y$ -intercept and one point: $a$ is the $y$ -intercept, then solve $a \cdot b^{x_{1}} = y_{1}$ for $b$ (positive root only).
From two points: $\frac{y _{2}}{y _{1}} = b^{x_{2} - x_{1}}$ cancels $a$ . Find $b$ first, then $a = \frac{y _{1}}{b ^{x_{1}}}$ .
Table with step $h$ and ratio $r$ : $b = r^{1/ h}$ , or write the model exactly as $y = a \cdot r^{x / h}$ .
Known asymptote $y = c$ : subtract $c$ from every $y$ -value, find $a$ and $b$ from $y - c = a \cdot b^{x}$ . The $y$ -intercept is then $a + c$ .
$a$ is the starting value and $b$ the factor per unit; $b - 1$ is the percent change per unit ( $b = 0.956$ means about 4.4% decrease).
Exponential regression fits $y = a \cdot b^{x}$ to all the data; check the residual pattern and keep predictions close to the data's range.

Solving Exponential Equations

Matching-base rule: if $b^{m} = b^{n}$ with $b > 0$ and $b \neq = 1$ , then $m = n$ , because $y = b^{x}$ never repeats an output.
Rewrite with the smallest common base, using negative exponents for reciprocals and rational exponents for roots: $\frac{1}{8} = 2^{- 3}$ , $39 = 3^{2/3}$ .
Replacing a base with a power multiplies the whole exponent: $(2^{3})^{x + 1} = 2^{3 x + 3}$ .
Isolate the power before matching. A coefficient is not part of the base: $3 (2)^{x} \neq = 6^{x}$ .
For a sum or difference of powers of one base, factor out the lowest power: $2^{x + 2} - 2^{x} = 2^{x} (2^{2} - 1)$ .
$b^{x} > 0$ always, so $b^{x}$ equal to a negative number or 0 has no solution; reject such factors.

Exponential Growth and Decay Applications

General model: $A (t) = A_{0} (1 + r)^{t}$ — growth if $r > 0$ , decay if $r < 0$ (or use $(1 - r)^{t}$ for a stated percentage decrease).
Half-life model: $A (t) = A_{0} (\frac{1}{2})^{t / h}$ , where $h$ is the half-life.
Doubling model: $A (t) = A_{0} (2)^{t / d}$ , where $d$ is the doubling time.
Matching bases: if $b^{m} = b^{n}$ then $m = n$ — rewrite both sides as powers of the same base to solve exactly.
Always keep $t$ and the rate's time period consistent (e.g. an annual rate needs $t$ measured in years).
Percent change per period = base − 1: a base of $0.92$ is 8% decay; a base of $1.075$ is 7.5% growth.

Unit 3: Discrete Functions

Sequences as Discrete Functions

A sequence is a discrete function: its domain is ${1, 2, 3, \dots}$ , and $t_{n}$ denotes the value of the $n$ th term.
An explicit formula gives $t_{n}$ directly in terms of $n$ — no previous terms required.
A recursive formula defines $t_{n}$ using earlier term(s), and must state a starting value such as $t_{1}$ .
Constant first differences $\Rightarrow$ arithmetic; constant ratio $\Rightarrow$ geometric; constant second differences $\Rightarrow$ quadratic-type.
Arithmetic recursion $t_{n} = t_{n - 1} + d$ matches $t_{n} = a + (n - 1) d$ ; geometric recursion $t_{n} = r t_{n - 1}$ matches $t_{n} = a r^{n - 1}$ .
The graph of a sequence is a set of separate points $(n, t_{n})$ , never a connected curve.

Arithmetic Sequences

Common difference: $d = t_{n} - t_{n - 1}$ (same value between every pair of consecutive terms).
Explicit formula: $t_{n} = a + (n - 1) d$ , where $a = t_{1}$ .
If $d > 0$ the sequence is increasing; if $d < 0$ it is decreasing; if $d = 0$ every term is identical.
Given two non-consecutive terms $t_{i}$ and $t_{j}$ : $d = \frac{t _{j} - t _{i}}{j - i}$ .
$t_{n} = d n + (a - d)$ is a linear function of $n$ with slope $d$ — arithmetic sequences are sampled lines.
Number of terms from $a$ to a last term $L$ : $n = \frac{L - a}{d} + 1$ .

Geometric Sequences

Common ratio: $r = \frac{t _{n}}{t _{n - 1}}$ (same value between every pair of consecutive terms).
Explicit formula: $t_{n} = a r^{n - 1}$ , where $a = t_{1}$ .
$∣ r ∣ > 1$ gives growth; $0 < ∣ r ∣ < 1$ gives decay toward 0; $r < 0$ makes the sequence alternate sign.
Given two non-consecutive terms $t_{i}$ and $t_{j}$ ( $j > i$ ): $r^{j - i} = \frac{t _{j}}{t _{i}}$ , so take the $(j - i)$ th root; if $j - i$ is even, both $\pm$ roots are valid ratios.
A $k %$ increase gives $r = 1 + \frac{k}{100}$ ; a $k %$ decrease gives $r = 1 - \frac{k}{100}$ .
To find $n$ , isolate $r^{n - 1}$ and match bases: $3 (2)^{n - 1} = 1536 \Rightarrow 2^{n - 1} = 2^{9} \Rightarrow n = 10 .$

Arithmetic Series

A series is the sum of a sequence's terms: $S_{n} = t_{1} + t_{2} + \dots + t_{n}$ .
Sum formula (needs last term): $S_{n} = \frac{n}{2} (a + t_{n})$ .
Sum formula (needs only $a$ and $d$ ): $S_{n} = \frac{n}{2} (2 a + (n - 1) d)$ .
Gauss's trick: pairing the first and last terms of an arithmetic series always gives the same sum, $a + t_{n}$ , and there are $\frac{n}{2}$ pairs.
Count terms before summing: solve $a + (n - 1) d = t_{n}$ for $n$ (the number of gaps plus 1).
A single term from partial sums: $t_{n} = S_{n} - S_{n - 1}$ ; and $1 + 2 + \dots + n = \frac{n ( n + 1 )}{2}$ .

Geometric Series

Geometric series sum: $S_{n} = \frac{a ( r ^{n} - 1 )}{r - 1}$ (equivalently $\frac{a ( 1 - r ^{n} )}{1 - r}$ ) for $r \neq = 1$ .
Special case: if $r = 1$ , every term equals $a$ , so $S_{n} = na$ .
Derivation idea: $S_{n} - r S_{n}$ cancels every term except the first and the shifted last term.
$a$ is the first term, $r$ is the common ratio, $n$ is how many terms are being added.
If the last term is given, find $n$ from $a r^{n - 1} = t_{n}$ first, or use $S_{n} = \frac{r t _{n} - a}{r - 1}$ .

Pascal's Triangle and the Binomial Theorem

Each entry in Pascal's triangle is the sum of the two entries diagonally above it; every row starts and ends with 1.
Row $n$ sums to $2^{n}$ .
Row $n$ gives the coefficients for expanding $(a + b)^{n}$ .
In each term of the expansion, the exponent of $a$ decreases from $n$ to $0$ while the exponent of $b$ increases from $0$ to $n$ — always summing to $n$ .
Recursive rule with rows and positions counted from 0: $t_{n, r} = t_{n - 1, r - 1} + t_{n - 1, r} .$
The term of $(a + b)^{n}$ containing $b^{r}$ is $t_{n, r} a^{n - r} b^{r}$ ; there are $n + 1$ terms in all.

Simple and Compound Interest

Simple interest: $I = P r t$ , and the final amount is $A = P (1 + r t)$ — grows arithmetically (linearly).
Compound interest: $A = P (1 + i)^{n}$ — grows geometrically, with common ratio $(1 + i)$ .
$i = \frac{r}{m}$ (rate per period) and $n = m t$ (number of periods), where $m$ = compounding periods per year.
Present value (solving backward): $P = A (1 + i)^{- n}$ .
Simple interest balances form an arithmetic sequence ( $d = P r$ ); compound balances form a geometric sequence ( $r = 1 + i$ ).
In $I = P r t$ , time must be in years: 9 months is $t = 0.75$ . Round money to the cent only at the end.

Simple Annuities

Ordinary simple annuity: equal payments $R$ at the end of each period, with the payment interval matching the compounding interval.
Future value: $F V = R \cdot \frac{( 1 + i ) ^{n} - 1}{i}$ .
Present value: $P V = R \cdot \frac{1 - ( 1 + i ) ^{- n}}{i}$ .
$n$ is the TOTAL number of payments, and $i$ is the interest rate per payment period — same matching rule as compound interest.
Both formulas come from a geometric series: the future value adds $R + R (1 + i) + \dots + R (1 + i)^{n - 1}$ .
Total paid $= n R$ . Loan interest $= n R - P V$ ; savings interest $= F V - n R$ .

Unit 4: Trigonometric Functions

Right Triangle Trigonometry

$sin θ = \frac{opposite}{hypotenuse}$ , $cos θ = \frac{adjacent}{hypotenuse}$ , $tan θ = \frac{opposite}{adjacent}$ .
Reciprocal ratios: $csc θ = \frac{1}{sin θ} = \frac{hyp}{opp}$ , $sec θ = \frac{1}{cos θ} = \frac{hyp}{adj}$ , $cot θ = \frac{1}{tan θ} = \frac{adj}{opp}$ .
To find a missing SIDE, set up the correct ratio and isolate the unknown.
To find a missing ANGLE, use the inverse function: $θ = sin^{- 1} (\dots)$ , $cos^{- 1} (\dots)$ , or $tan^{- 1} (\dots)$ .
Angle of elevation (looking up) = angle of depression (looking back down) between the same two points, since horizontal lines are parallel.
Two-triangle problems: solve for the shared side first, then add or subtract, rounding only at the end.

Special Triangles and Exact Values

45-45-90 triangle side ratio: $1 : 1 : 2$ .
30-60-90 triangle side ratio: $1 : 3 : 2$ (shortest side opposite $30°$ , longest leg opposite $60°$ , hypotenuse opposite $90°$ ).
$sin 30° = cos 60° = \frac{1}{2}$ ; $sin 60° = cos 30° = \frac{3}{2}$ ; $sin 45° = cos 45° = \frac{2}{2}$ .
$tan 30° = \frac{3}{3}$ , $tan 45° = 1$ , $tan 60° = 3$ ; $tan 90°$ is undefined.
On the unit circle, the point at angle $θ$ has coordinates $(cos θ, sin θ)$ .
$sin^{2} θ$ means $(sin θ)^{2}$ ; reciprocal exact values come from flipping, e.g. $csc 30° = 2$ , $sec 45° = 2$ .

Angles from 0° to 360° and the CAST Rule

Reference angle: the acute angle between the terminal arm and the $x$ -axis; its trig ratios share the SAME magnitude as the original angle's.
For a point $P (x, y)$ on the terminal arm: $r = x^{2} + y^{2}$ , $sin θ = \frac{y}{r}$ , $cos θ = \frac{x}{r}$ , $tan θ = \frac{y}{x}$ .
CAST rule: QI all positive, QII sine positive, QIII tangent positive, QIV cosine positive (everything else in that quadrant is negative).
A negative angle $- θ$ is coterminal with $360° - θ$ , and has identical trig ratios.
Reciprocal ratios: $csc θ = \frac{1}{sin θ}$ , $sec θ = \frac{1}{cos θ}$ , $cot θ = \frac{1}{tan θ}$ — same sign as the original ratio.
Solving $sin θ$ , $cos θ$ or $tan θ = k$ (with $0 < ∣ k ∣ < 1$ for sine or cosine, or any $k \neq = 0$ for tangent) on $0° \leq θ < 360°$ gives TWO angles: same reference angle, two quadrants chosen by CAST.

The Sine Law

Sine law: $\frac{a}{sin A} = \frac{b}{sin B} = \frac{c}{sin C}$ , or flipped for angles: $\frac{sin A}{a} = \frac{sin B}{b} = \frac{sin C}{c}$ .
Each side is paired with the angle directly opposite it.
Use the sine law when you know an angle and its opposite side (ASA, AAS, or SSA).
Angles in any triangle always sum to $180°$ .
It comes from writing the same altitude two ways: $h = b sin A = a sin B$ .
3D problems: split into flat triangles that share a side; solve the one with enough information first.

The Ambiguous Case of the Sine Law (SSA)

SSA (two sides and a non-included angle) can produce 0, 1, or 2 triangles — it's the only ambiguous case.
The cause: $sin θ = sin (180° - θ)$ , so one sine value fits an acute AND an obtuse angle.
Test height: $h = b sin A$ (with $A$ acute, $b$ adjacent to $A$ , $a$ opposite $A$ ).
$a < h$ : no triangle. $a = h$ : one (right) triangle. $h < a < b$ : two triangles. $a \geq b$ : one triangle.
Given angle $90°$ or more: one triangle if $a > b$ , none if $a \leq b$ .
In the two-triangle case, the second angle is $B_{2} = 180° - B_{1}$ ; check it still leaves $A + B_{2} < 180°$ .

The Cosine Law

Cosine law: $c^{2} = a^{2} + b^{2} - 2 ab cos C$ , where $C$ is included between sides $a$ and $b$ .
Use it for SAS (find the missing side) or SSS (find any angle).
Rearranged to solve for an angle: $cos C = \frac{a ^{2} + b ^{2} - c ^{2}}{2 ab}$ .
A negative value of $cos C$ means $C$ is obtuse — this is expected, not an error.
It comes from Pythagoras plus $sin^{2} C + cos^{2} C = 1$ ; at $C = 90°$ it reduces to $c^{2} = a^{2} + b^{2}$ .
Solving a whole triangle: cosine law first (largest angle in SSS), then the sine law for the smaller angles, then the angle sum.

Trigonometric Identities

Pythagorean identity: $sin^{2} θ + cos^{2} θ = 1$ for every angle $θ$ .
Quotient identity: $tan θ = \frac{sin θ}{cos θ}$ .
Rearranged Pythagorean forms: $sin^{2} θ = 1 - cos^{2} θ$ and $cos^{2} θ = 1 - sin^{2} θ$ .
To prove an identity, simplify LS and RS independently until they match — never operate across the equals sign.
One angle where LS and RS differ disproves a statement; checking a few angles never proves one.
Useful moves: write $tan$ as $\frac{s i n}{c o s}$ , combine fractions over a common denominator, factor, and replace $sin^{2} θ + cos^{2} θ$ with $1$ .

Periodic Functions: Period, Amplitude, and Axis

Period: the horizontal distance for one complete repeating cycle.
Axis (midline): $\frac{max + min}{2}$ — the horizontal centre line of the graph.
Amplitude: $\frac{max - min}{2}$ — always reported as a positive distance.
A function is periodic if its values repeat in a fixed cycle forever.
Because $f (x + p) = f (x)$ , subtracting whole periods predicts future values: $f (x + n p) = f (x)$ .
Checks: $max = axis + amplitude$ and $min = axis - amplitude$ .

Graphs of y = sin x and y = cos x — Transformations

Parent curves: $y = sin x$ and $y = cos x$ both have amplitude $1$ , period $360°$ , and axis $y = 0$ .
In $y = a sin (k (x - d)) + c$ : $∣ a ∣$ = amplitude, $\frac{360°}{∣ k ∣}$ = period, $d$ = phase shift (right if positive), $c$ = axis.
A negative $a$ reflects the graph vertically (upside down) but leaves the amplitude, period, and axis unchanged.
$sin x$ starts at the axis and rises; $cos x$ starts at its maximum — they're the same shape, just $90°$ apart.
Order: stretches and reflections first, then translations. Mapping rule: $(x, y) \to (\frac{x}{k} + d, a y + c)$ .
Factor $k$ out before reading $d$ : $sin (2 x - 60°) = sin (2 (x - 30°))$ has $d = 30°$ .

Modelling with Sinusoidal Functions

Axis: $c = \frac{max + min}{2}$ . Amplitude: $a = \frac{max - min}{2}$ . Both always come first.
$k = \frac{360°}{period}$ .
If a scenario starts at a maximum, model with $+ cos$ ; if it starts at a minimum, model with $- cos$ ; if it starts at the axis rising, use $+ sin$ .
Once $a, k, c$ are set from the max/min/period, only the starting behaviour decides which parent function and sign to use — often avoiding the need for a $d$ term entirely.
In context: $c$ is the average level (a Ferris wheel's centre height), $∣ a ∣$ is half the range (the radius), the period is the time for one cycle, and $d$ is the time of the chosen starting feature.
To find WHEN a model reaches a value: set it equal, isolate the trig ratio, use the related angle and CAST (or read the graph), then add whole periods for later cycles. Each value between max and min occurs twice per cycle.