4.1 — Solving Exponential Equations (Same-Base Method)
Rewrite both sides with a common base, then equate exponents; recognise when the same-base technique applies.
Solve \( 4^{x+1} = 8^{x-2} \): rewrite as \( (2^2)^{x+1}=(2^3)^{x-2} \), so \( 2^{2x+2}=2^{3x-6} \). Equate exponents: \( 2x+2=3x-6 \), giving \( x=8 \). When no common base exists (e.g. \( 2^x=7 \)), logarithms are required — that's MHF4U.
4.2 — Exponential Growth Models
Use \( A(t)=A_0(1+r)^t \) and \( A(t)=A_0\,b^{t/T} \) for population growth, viral spread, and continuous appreciation.
A culture starts with 500 cells and doubles every 3 hours. After \( t \) hours: \( A(t)=500\cdot 2^{t/3} \). After 12 hours: \( A(12)=500\cdot 2^4=8000 \) cells. After 1 day: \( A(24)=500\cdot 2^8=128{,}000 \). The form \( b^{t/T} \) is read “b raised to (time per period)” — the doubling time form.
4.3 — Exponential Decay & Half-Life
Use \( A(t)=A_0(\tfrac{1}{2})^{t/h} \) for radioactive decay, drug elimination, and cooling; relate half-life to the decay constant.
Carbon-14 has a half-life of 5730 years. A 100 g sample after \( t \) years: \( A(t)=100(\tfrac{1}{2})^{t/5730} \). After 11{,}460 years: \( A=100(\tfrac{1}{2})^2=25 \) g. Equivalently: \( A(t)=100(0.5)^{t/5730} \). Decay models always have base \( 0
4.4 — Compound Interest as an Exponential Application
Apply \( A=P(1+i)^n \) where \( i \) is the rate per compounding period and \( n \) is the number of compounding periods; preview the formal treatment in Ch 6.
$1000 invested at 6% per annum compounded quarterly for 5 years: \( i = 0.06/4=0.015 \), \( n=5\cdot 4=20 \). \( A=1000(1.015)^{20}\approx \$1{,}346.86 \). Exponential growth at work — and the foundation for the more rigorous treatment of present value, future value, and annuities in Chapter 6.
5.1 — Sequences: Recursive vs. General-Term Definitions
Define a sequence as a function with domain \( \mathbb{N} \); express explicitly (\( t_n = \) formula in \( n \)) or recursively (\( t_n=t_{n-1}+\dots \)).
The Fibonacci sequence \( 1,1,2,3,5,8,\ldots \) is recursive: \( t_n=t_{n-1}+t_{n-2}, \, t_1=t_2=1 \). The same sequence has no simple closed-form. By contrast \( t_n=2n+1 \) gives \( 3,5,7,9,\ldots \) — both forms describe the sequence; they emphasise different patterns.
5.2 — Arithmetic Sequences
Identify a constant common difference \( d \); use \( t_n = a + (n-1)d \) where \( a=t_1 \).
Sequence: \( 7, 11, 15, 19, \ldots \); \( a=7, d=4 \). The 25th term is \( t_{25}=7+(25-1)(4)=7+96=103 \). To find which term equals 167: \( 167=7+(n-1)(4) \) gives \( n=41 \). The constant first difference is the function-class signature of an arithmetic (linear) sequence.
5.3 — Geometric Sequences
Identify a constant common ratio \( r \); use \( t_n = a\,r^{n-1} \).
Sequence: \( 3, 6, 12, 24, \ldots \); \( a=3, r=2 \). The 10th term is \( t_{10}=3\cdot 2^9=3\cdot 512=1536 \). Geometric sequences are discrete exponential functions — the link between Strands B and C.
5.4 — Arithmetic Series \( S_n \)
Sum the first \( n \) terms of an arithmetic sequence: \( S_n = \dfrac{n}{2}\,(2a+(n-1)d) = \dfrac{n}{2}(t_1+t_n) \).
Sum the first 50 odd numbers: \( a=1, d=2 \). \( S_{50}=\tfrac{50}{2}(2(1)+49(2))=25(100)=2500 \). The famous “Gauss trick” — pair the first and last terms — produces this same formula.
5.5 — Geometric Series & Sigma Notation
Sum the first \( n \) terms of a geometric sequence: \( S_n = \dfrac{a(r^n - 1)}{r-1} \) (\( r\ne 1 \)); read and write series in sigma notation.
Sum: \( 2+6+18+\ldots+2(3)^9 \). Here \( a=2, r=3, n=10 \). \( S_{10}=\dfrac{2(3^{10}-1)}{3-1}=\dfrac{2(59048)}{2}=59048 \). In sigma notation: \( \displaystyle\sum_{k=1}^{10} 2\cdot 3^{k-1} \).
6.1 — Compound Interest: Future Value & Present Value
Apply \( A=P(1+i)^n \); solve for \( P \) (present value) by rearranging: \( P=A(1+i)^{-n} \).
How much to invest now to have \$10{,}000 in 8 years at 5%/yr compounded annually? \( P=10000(1.05)^{-8}\approx \$6{,}768.39 \). Present value answers “what's it worth today?”; future value answers “what will it grow to?”.
6.2 — Compounding Periods & Effective Rates
Convert a nominal annual rate to a periodic rate \( i = j/m \) for \( m \) compounding periods per year; compare effective annual yields.
A 6% annual rate compounded monthly: \( i=0.06/12=0.005 \); after 1 year, factor is \( (1.005)^{12}\approx 1.0617 \), so the effective annual rate is \( \approx 6.17\% \). More frequent compounding → higher effective yield.
6.3 — Future Value of an Ordinary Annuity
Use \( FV = R\,\dfrac{(1+i)^n - 1}{i} \) for equal periodic deposits \( R \) made at the end of each period.
Save \$200 monthly for 30 years at 6%/yr compounded monthly: \( i=0.005, n=360, R=200 \). \( FV=200\cdot\dfrac{1.005^{360}-1}{0.005}\approx \$200{,}903.01 \). Notice the formula is a finite geometric series — Ch 5 in disguise.
6.4 — Present Value of an Ordinary Annuity
Use \( PV = R\,\dfrac{1-(1+i)^{-n}}{i} \) to find the lump sum equivalent to a stream of equal future payments.
A retiree wants \$2{,}000/month for 25 years at 4%/yr compounded monthly: \( i=0.04/12 \approx 0.00333, n=300 \). \( PV \approx 2000\cdot \dfrac{1-(1.00333)^{-300}}{0.00333}\approx \$378{,}824 \). The required lump-sum invested today.
6.5 — Mortgages & Amortization
Apply PV-of-annuity to find mortgage payment \( R \); construct an amortization schedule (interest vs. principal).
A \$300{,}000 mortgage at 5%/yr compounded semi-annually, 25-year amortization: convert to monthly equivalent rate, solve \( 300{,}000=R\cdot\dfrac{1-(1+i)^{-300}}{i} \) for \( R\approx \$1{,}744.81 \) per month. Early payments are mostly interest; later payments are mostly principal.
7.1 — Primary & Reciprocal Trigonometric Ratios
Define \( \sin\theta = \tfrac{\text{opp}}{\text{hyp}}, \cos\theta = \tfrac{\text{adj}}{\text{hyp}}, \tan\theta = \tfrac{\text{opp}}{\text{adj}} \) and the reciprocals \( \csc, \sec, \cot \) for acute angles.
In a right triangle with opposite 5, adjacent 12, hypotenuse 13: \( \sin\theta=\tfrac{5}{13}, \cos\theta=\tfrac{12}{13}, \tan\theta=\tfrac{5}{12} \) and \( \csc\theta=\tfrac{13}{5}, \sec\theta=\tfrac{13}{12}, \cot\theta=\tfrac{12}{5} \). Reciprocal ratios extend the toolkit beyond the three primary ratios.
7.2 — Trigonometric Ratios of Any Angle (0°–360°)
Use the unit circle and the CAST rule to determine the sign of each trig ratio; find related/reference angles.
For \( \theta \) in standard position with terminal arm passing through \( (-3, 4) \): \( r=5 \), so \( \sin\theta=4/5, \cos\theta=-3/5, \tan\theta=-4/3 \). The reference angle is \( \tan^{-1}(4/3)\approx 53.13° \); the principal angle is \( 180°-53.13°=126.87° \). Quadrant II: only sine positive (CAST).
7.3 — Special Angles & Exact Values
Memorise the exact ratios for 0°, 30°, 45°, 60°, 90° and their multiples in standard position.
Key exact values: \( \sin 30°=\tfrac{1}{2}, \cos 30°=\tfrac{\sqrt{3}}{2}, \tan 30°=\tfrac{1}{\sqrt{3}} \); \( \sin 45°=\cos 45°=\tfrac{\sqrt{2}}{2}, \tan 45°=1 \); \( \sin 60°=\tfrac{\sqrt{3}}{2}, \cos 60°=\tfrac{1}{2}, \tan 60°=\sqrt{3} \). Derived from 30-60-90 and 45-45-90 reference triangles.
7.4 — The Sine Law & the Ambiguous Case
Apply \( \dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C} \) to non-right triangles; recognise SSA situations producing 0, 1, or 2 triangles.
In \( \triangle ABC \): \( a=10, b=14, A=35° \). By sine law: \( \sin B=\dfrac{14\sin 35°}{10}\approx 0.803 \), so \( B\approx 53.4° \) or \( B\approx 126.6° \) — two triangles satisfy the data. Ambiguous case (SSA): always check whether the supplementary angle also gives a valid triangle (i.e. \( A+B<180° \)).
7.5 — The Cosine Law & 2D/3D Applications
Use \( c^2 = a^2 + b^2 - 2ab\cos C \) for SAS and SSS triangles; apply both laws in surveying, navigation, and 3D problems.
In \( \triangle ABC \): \( a=8, b=11, C=42° \). Then \( c^2=64+121-2(8)(11)\cos 42°=185-130.79\approx 54.21 \), so \( c\approx 7.36 \). Cosine law is needed when sine law has insufficient information (no opposite-pair).
8.1 — Graphs of \( y=\sin x \) and \( y=\cos x \)
Sketch the parent sinusoidal functions in degree mode; identify period (360°), amplitude (1), domain (\(\mathbb{R}\)), range (\([-1,1]\)).
The sine function passes through (0,0), peaks at (90°,1), returns to 0 at (180°,0), reaches a minimum at (270°,-1), then back to 0 at (360°,0). Cosine is sine shifted left by 90°: \( \cos x = \sin(x+90°) \). Both repeat every 360° — they are periodic.
8.2 — Amplitude, Period, & Vertical Shift
Analyse \( y = a\sin(kx) + c \): amplitude \( |a| \), period \( \dfrac{360°}{|k|} \), vertical shift \( c \), midline \( y=c \).
For \( y=3\sin(2x)+1 \): amplitude 3, period \( 360°/2=180° \), midline \( y=1 \), maximum \( y=4 \), minimum \( y=-2 \). Stretch in the y-direction (amplitude) and compression in the x-direction (period) reshape the parent sine curve.
8.3 — Phase Shift & the General Sinusoidal Form
Analyse the full form \( y = a\sin(k(x-d)) + c \): the parameter \( d \) is a horizontal (phase) shift; combine with all other transformations.
For \( y=2\sin(3(x-30°))-1 \): amplitude 2, period 120°, phase shift 30° right, midline \( y=-1 \). Always factor out \( k \) inside the argument before reading off the phase shift: \( \sin(3x-90°)=\sin(3(x-30°)) \).
8.4 — Modeling Periodic Phenomena
Build sinusoidal models from data: tides, daylight hours, sound waves, Ferris-wheel height, pendulum motion.
Tide example: water depth varies between 1.5 m (low) and 7.5 m (high), with high tide at \( t=3 \) h and a 12-hour period. Midline \( c=4.5 \), amplitude \( a=3 \), period 12 → \( k=360°/12=30°/\text{h} \). Cosine fits cleanly with peak at \( t=3 \): \( h(t)=3\cos(30°(t-3))+4.5 \).
8.5 — Solving Sinusoidal Equations Graphically
Use a graph or table to solve equations of the form \( a\sin(k(x-d))+c = K \) on a given interval; interpret in context.
From the tide model \( h(t)=3\cos(30°(t-3))+4.5 \), find when \( h=6 \) m. Solve \( 3\cos(30°(t-3))=1.5 \), so \( \cos(30°(t-3))=0.5 \), giving \( 30°(t-3)=\pm 60° \), so \( t=1 \) h or \( t=5 \) h within one period. Algebraic solution methods are studied formally in MHF4U.
B.1 — Average vs. Instantaneous Rate of Change (MHF4U Preview)
For any function studied in MCR3U (quadratic, exponential, sinusoidal), the average rate of change over \([a,b]\) is \( \frac{f(b)-f(a)}{b-a} \) — the slope of the secant line. As \( b\to a \), the secant approaches the tangent, giving the instantaneous rate of change. This limit concept is the gateway to calculus and is developed formally in MHF4U and MCV4U.
For \( f(x)=x^2 \), the average rate of change on \([2,2+h]\) is \( \frac{(2+h)^2 - 4}{h} = 4+h \). As \( h\to 0 \), the instantaneous rate at \( x=2 \) is \( 4 \). The same secant-to-tangent reasoning applies to exponential growth (population doubling rate) and sinusoidal motion (instantaneous velocity of a Ferris wheel rider). Mastering this difference quotient now makes MHF4U's introduction to derivatives feel natural.
B.2 — Comprehensive MCR3U Exam Review (61 Questions from Past Tests)
A full walkthrough of 61 questions drawn from past MCR3U unit tests and final exams, covering all four strands: characteristics of functions, exponential functions, discrete/financial mathematics, and trigonometry. Use this to consolidate before the final and to identify any lingering misconceptions before MHF4U.
Common misconceptions to watch for: (1) confusing \( (f^{-1}(x)) \) with \( \frac{1}{f(x)} \); (2) applying transformations in the wrong order — remember inside the bracket reverses, outside applies directly; (3) treating \( (a+b)^2 = a^2+b^2 \) (it isn't); (4) forgetting the ambiguous case in sine law when given SSA; (5) using simple instead of compound interest formula on annuity problems. This review surfaces and corrects all of these.