Grade 11 · MCR3U · Functions

MCR3U Formula Sheet

Q: Do you get a formula sheet on the MCR3U exam?

That is up to your teacher. Some hand one out, some let you bring your own page, some give none. Ask early, and learn the formulas either way: knowing which one to use is half the question.

Q: Which MCR3U formulas should I memorize?

The quadratic formula, the exponent laws, the sine and cosine laws, the special angle values, and the sequence and series formulas. They show up in almost every question in their units.

Every formula from Grade 11 Functions, grouped by unit, in the order you learn them. Print the PDF for your binder or keep this page open while you practise. No signup needed.

Download the PDF MCR3U practice exam

MCR3U Formula Sheet · Grade 11 FunctionsOntario curriculum

Functions & Transformations

Function notation: $y = f(x)$
Each $x$ in the domain gives exactly one $y$ .
Transformed function: $y = af\big[k(x - d)\big] + c$
Mapping rule: $(x,\ y) \rightarrow \left(\frac{x}{k} + d,\ ay + c\right)$
Stretch or reflect first, then translate.
Inverse: $f(a) = b \iff f^{-1}(b) = a$
Swap $x$ and $y$ , then solve for $y$ .
Parent functions: $y = x^2 \qquad y = \frac{1}{x} \qquad y = \sqrt{x} \qquad y = |x|$

Rational Expressions

Multiplying radicals: $\sqrt{a}\,\sqrt{b} = \sqrt{ab}$
For $a, b \geq 0$ .
Dividing radicals: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$
For $a \geq 0$ , $b > 0$ .
Conjugates: $\left(\sqrt{a} + \sqrt{b}\right)\left(\sqrt{a} - \sqrt{b}\right) = a - b$
Multiplying: $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$
Restrictions: $b, d \neq 0$ .
Dividing: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$
Restrictions: $b, c, d \neq 0$ .
Adding: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$
Use the lowest common denominator when there is one.

Quadratic Functions

Standard form: $y = ax^2 + bx + c$
Vertex form: $y = a(x - h)^2 + k$
Vertex $(h, k)$ , axis of symmetry $x = h$ .
Factored form: $y = a(x - r)(x - s)$
Zeros $r$ and $s$ , axis of symmetry $x = \frac{r + s}{2}$ .
Vertex from standard form: $x = -\frac{b}{2a}$
Quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Discriminant: $D = b^2 - 4ac$
$D > 0$ : two real roots. $D = 0$ : one. $D < 0$ : none.

Exponential Functions

Product and quotient laws: $a^m \cdot a^n = a^{m+n} \qquad \frac{a^m}{a^n} = a^{m-n}$
Power laws: $\left(a^m\right)^n = a^{mn} \qquad (ab)^n = a^n b^n$
Zero and negative exponents: $a^0 = 1 \qquad a^{-n} = \frac{1}{a^n}$
For $a \neq 0$ .
Rational exponents: $a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m$
Transformed exponential: $y = a\,b^{k(x - d)} + c$
Growth and decay: $A = A_0(1 \pm r)^t$
$r$ is the rate per period, $t$ the number of periods.
Half-life: $A = A_0\left(\frac{1}{2}\right)^{\frac{t}{h}}$
$h$ is the half-life.
Doubling: $A = A_0(2)^{\frac{t}{d}}$
$d$ is the doubling time.

Trigonometric Ratios

Primary ratios: $\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$
Reciprocal ratios: $\csc\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$
Angle in standard position: $\sin\theta = \frac{y}{r} \qquad \cos\theta = \frac{x}{r} \qquad \tan\theta = \frac{y}{x}$
For a point $(x, y)$ on the terminal arm, $r = \sqrt{x^2 + y^2}$ .
Special angles: $\sin 30^\circ = \tfrac{1}{2} \qquad \sin 45^\circ = \tfrac{\sqrt{2}}{2} \qquad \sin 60^\circ = \tfrac{\sqrt{3}}{2}$
And $\cos\theta = \sin(90^\circ - \theta)$ .
Sine law: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$
Cosine law: $a^2 = b^2 + c^2 - 2bc\cos A$
Ambiguous case: $\sin\theta = \sin\left(180^\circ - \theta\right)$
Given two sides and an angle opposite one of them, check both angles.
Quotient identity: $\tan\theta = \frac{\sin\theta}{\cos\theta}$
Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$

Trigonometric Functions

Sinusoidal function: $y = a\sin\big[k(x - d)\big] + c \qquad y = a\cos\big[k(x - d)\big] + c$
Period: $\text{period} = \frac{360^\circ}{|k|}$
Amplitude: $|a| = \frac{\max - \min}{2}$
Axis of the curve: $y = c = \frac{\max + \min}{2}$
Mapping rule: $(x,\ y) \rightarrow \left(\frac{x}{k} + d,\ ay + c\right)$

Discrete Functions

Arithmetic sequence: $t_n = a + (n - 1)d$
Geometric sequence: $t_n = ar^{n-1}$
Arithmetic series: $S_n = \frac{n}{2}\big[2a + (n - 1)d\big] = \frac{n}{2}\left(t_1 + t_n\right)$
Geometric series: $S_n = \frac{a\left(r^n - 1\right)}{r - 1}$
For $r \neq 1$ .
Pascal's triangle: $t_{n,r} = t_{n-1,\,r-1} + t_{n-1,\,r}$
Row $n$ gives the coefficients of $(a + b)^n$ .
Simple interest: $I = Prt \qquad A = P + I$
Compound interest: $A = P(1 + i)^n$
$i$ is the rate per compounding period, $n$ the number of periods.
Present value: $PV = A(1 + i)^{-n}$
Future value of an annuity: $FV = \frac{R\big[(1 + i)^n - 1\big]}{i}$
Present value of an annuity: $PV = \frac{R\big[1 - (1 + i)^{-n}\big]}{i}$

MCR3U Formula Sheet FAQ

Do you get a formula sheet on the MCR3U exam?+

That is up to your teacher. Some hand one out, some let you bring your own page, some give none. Ask early, and learn the formulas either way: knowing which one to use is half the question.

Which MCR3U formulas should I memorize?+

The quadratic formula, the exponent laws, the sine and cosine laws, the special angle values, and the sequence and series formulas. They show up in almost every question in their units.

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