Final Examination

At a glance

Individual · three hours, in the examination period · written, with a calculator and one formula sheet supplied · all four strands, weighted roughly as the course spent its time

What it is for

Everything else this semester was worked in groups, at the boards, with time to look things up and somebody beside you. This is the one piece of evidence that is unambiguously yours. It is also the closest thing in this course to the assessments Grade 12 and first-year mathematics will use constantly, met here where I am the one marking it.

What is on it

Part	Roughly	What it asks you to do
A. Functions and algebra	30%	Function notation, domain and range, inverses, transformations; simplify polynomial and rational expressions and state restrictions; solve quadratics and find where a line meets a curve
B. Exponential functions	20%	Rational exponents and exponent laws; graph and transform exponential functions; find an equation from a graph; solve growth and decay problems
C. Sequences, series, and finance	25%	Recursive and explicit forms; arithmetic and geometric sequences and series; simple and compound interest, annuities and loans
D. Trigonometry	25%	Exact ratios and the reciprocal ratios; prove identities; the sine and cosine laws; sinusoidal graphs, equations from data, and problems

Every part asks you to show the route. A bare answer earns almost nothing; a clearly wrong number reached by correct reasoning earns most of the marks.

What to expect, precisely

• Restrictions are marked. Simplifying a rational expression without stating what $x$ cannot be is an incomplete answer, every time — Equivalent Algebraic Expressions is where that habit was built.
• Identity proofs are marked on layout as much as on algebra: one side only, each line following from the one above. The Other Three Ratios has the method.
• One question gives you data — a table of measurements — and asks for a sinusoidal model, a prediction, and an honest sentence about how far ahead you would trust it. That is Catching a Wave under exam conditions.
• Financial questions may need a TVM solver. You will be asked to state what you entered as well as what came out, because the entries are where the thinking is. Interest as a Sequence covers the two conventions that cause most errors.
• Exact values where exact values exist. $\frac{\sqrt{3}}{2}$ is an answer; 0.866 is a rounding of one.

How to prepare

1. Redo problems, do not reread them. Five from each unit’s practice set, cold, on paper, with only the formula sheet.
2. Rebuild the connections. Arithmetic sequence, linear function, simple interest are one idea; geometric, exponential, compound interest are another. Draw that map from memory — the gaps are your study list.
3. Re-derive rather than memorise. The period of $y=a\sin (k(x-d))+c$ is $\frac{360^{\circ }}{|k|}$ because $k$ compresses horizontally. Knowing why survives pressure; knowing the formula alone does not.
4. Practise the proofs out loud. If you cannot say the first move before you write it, you are hoping rather than working.
5. Bring questions to the review classes. This page is what is on the examination.

In the three hours

Budget by the weightings and hold to them — twenty minutes over on Part A costs more than it gains. If a question stalls, write down what you know about it and what you intended to do; a described method earns marks and a blank space earns none.

How this is assessed

Against the same expectations as everything else. Per How Marks Work, this examination is part of the final 30% of the course mark alongside The Functions Symposium, so that neither one afternoon nor one project decides your grade alone.

Curriculum connection

A3.2

verify, through investigation with and without technology, that $\sqrt{ab}=\sqrt{a}\times \sqrt{b}$, $a\ge 0$, $b\ge 0$, and use this relationship to simplify radicals (e.g., $\sqrt{24}$) and radical expressions obtained by adding, subtracting, and multiplying [e.g., $(2+\sqrt{6})(3-\sqrt{12})$]

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A3.3

simplify rational expressions by adding, subtracting, multiplying, and dividing, and state the restrictions on the variable values Sample problem: Simplify $\frac{2x}{4x^2+6x}-\frac{3}{2x+3}$, and state the restrictions on the variable.

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B1.4

determine, through investigation, and describe key properties relating to domain and range, intercepts, increasing/decreasing intervals, and asymptotes (e.g., the domain is the set of real numbers; the range is the set of positive real numbers; the function either increases or decreases throughout its domain) for exponential functions represented in a variety of ways [e.g., tables of values, mapping diagrams, graphs, equations of the form $f(x)=a^x$ $(a>0,a\ne 1)$, function machines] Sample problem: Graph $f(x)=2^x$, $g(x)=3^x$, and $h(x)=0.5^x$ on the same set of axes. Make comparisons between the graphs, and explain the relationship between the y-intercepts.

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B2.3

sketch graphs of $y=af(k(x-d))+c$ by applying one or more transformations to the graph of $f(x)=a^x$ $(a>0,a\ne 1)$, and state the domain and range of the transformed functions Sample problem: Transform the graph of $f(x)=3^x$ to sketch $g(x)=3^{-(x+1)}-2$, and state the domain and range of each function.

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C1.2

determine and describe (e.g., in words; using flow charts) a recursive procedure for generating a sequence, given the initial terms (e.g., 1, 3, 6, 10, 15, 21, …), and represent sequences as discrete functions in a variety of ways (e.g., tables of values, graphs)

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D1.2

determine the values of the sine, cosine, and tangent of angles from $0^{\circ }$ to $360^{\circ }$, through investigation using a variety of tools (e.g., dynamic geometry software, graphing tools) and strategies (e.g., applying the unit circle; examining angles related to special angles)

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D1.5

prove simple trigonometric identities, using the Pythagorean identity ${\sin }^{2}x+{\cos }^{2}x=1$; the quotient identity $\tan x=\frac{\sin x}{\cos x}$; and the reciprocal identities $\sec x=\frac{1}{\cos x}$, $\csc x=\frac{1}{\sin x}$, and $\cot x=\frac{1}{\tan x}$ Sample problem: Prove that $1-{\cos }^{2}x=\sin x\cos x\tan x$.

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