Grade 12 · MHF4U · Advanced Functions

MHF4U Formula Sheet

Every formula from Advanced 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.

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MHF4U Formula Sheet · Grade 12 Advanced FunctionsOntario curriculum

Polynomial Functions

Polynomial function: $f(x) = a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$
Degree $n$ , leading coefficient $a_n$ .
Transformed power function: $y = a\big[k(x - d)\big]^n + c$
Even function: $f(-x) = f(x)$
Symmetric about the $y$ -axis.
Odd function: $f(-x) = -f(x)$
Symmetric about the origin.
Average rate of change: $\frac{\Delta y}{\Delta x} = \frac{f(b) - f(a)}{b - a}$
Instantaneous rate of change (estimate): $\frac{f(a + h) - f(a)}{h}$
With $h$ very small.
Finite differences: $\Delta^n y = a \cdot n!$
For degree $n$ and equal $x$ steps of 1, the $n$ th differences are constant.

Polynomial Equations

Division statement: $P(x) = D(x)\,Q(x) + R(x)$
Remainder theorem: $P(x) \div (x - a) \text{ leaves remainder } P(a)$
Dividing by $(ax - b)$ leaves $P\left(\frac{b}{a}\right)$ .
Factor theorem: $(x - a) \text{ is a factor of } P(x) \iff P(a) = 0$
Rational zero test: $x = \frac{b}{a}$
$b$ divides the constant term, $a$ divides the leading coefficient.
Sum of cubes: $a^3 + b^3 = (a + b)\left(a^2 - ab + b^2\right)$
Difference of cubes: $a^3 - b^3 = (a - b)\left(a^2 + ab + b^2\right)$

Rational Functions

Reciprocal function: $y = \frac{1}{f(x)}$
Vertical asymptotes where $f(x) = 0$ ; invariant points where $f(x) = \pm 1$ .
Linear over linear: $y = \frac{ax + b}{cx + d}$
Vertical asymptote $x = -\frac{d}{c}$ , horizontal asymptote $y = \frac{a}{c}$ .
Horizontal asymptote: $\deg N < \deg D:\ y = 0 \qquad \deg N = \deg D:\ y = \frac{a}{b}$
$a$ and $b$ are the leading coefficients of $N$ and $D$ . When $\deg N = \deg D + 1$ there is an oblique asymptote instead: divide.
Hole: $f(x) = \frac{(x - a)\,g(x)}{(x - a)\,h(x)}$
A factor that cancels gives a hole at $x = a$ , not an asymptote.

Trigonometric Ratios

Radians and degrees: $\pi \text{ rad} = 180^\circ$
Arc length: $a = r\theta$
$\theta$ in radians.
Special angles: $\sin\tfrac{\pi}{6} = \tfrac{1}{2} \qquad \sin\tfrac{\pi}{4} = \tfrac{\sqrt{2}}{2} \qquad \sin\tfrac{\pi}{3} = \tfrac{\sqrt{3}}{2}$
Reciprocal ratios: $\csc x = \frac{1}{\sin x} \qquad \sec x = \frac{1}{\cos x} \qquad \cot x = \frac{1}{\tan x}$
Quotient identity: $\tan x = \frac{\sin x}{\cos x}$
Pythagorean identity: $\sin^2 x + \cos^2 x = 1$
Pythagorean identities, tan and cot: $1 + \tan^2 x = \sec^2 x \qquad 1 + \cot^2 x = \csc^2 x$
Co-function identities: $\sin x = \cos\left(\frac{\pi}{2} - x\right) \qquad \cos x = \sin\left(\frac{\pi}{2} - x\right)$
Shift by a quarter turn: $\sin\left(x + \frac{\pi}{2}\right) = \cos x \qquad \cos\left(x + \frac{\pi}{2}\right) = -\sin x$
Compound angle, sine: $\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y$
Compound angle, cosine: $\cos(x \pm y) = \cos x\cos y \mp \sin x\sin y$
Double angle: $\sin 2x = 2\sin x\cos x \qquad \tan 2x = \frac{2\tan x}{1 - \tan^2 x}$
Double angle, cosine: $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$

Trigonometric Functions

Sinusoidal function: $y = a\sin\big[k(x - d)\big] + c \qquad y = a\cos\big[k(x - d)\big] + c$
Period: $\text{period} = \frac{2\pi}{|k|}$
For $y = \tan kx$ the period is $\frac{\pi}{|k|}$ .
Amplitude and axis: $|a| = \frac{\max - \min}{2} \qquad c = \frac{\max + \min}{2}$
Asymptotes of tangent: $y = \tan x: \quad x = \frac{\pi}{2} + n\pi,\ n \in \mathbb{Z}$

Exponential & Logarithmic Functions

Definition of a logarithm: $y = b^x \iff x = \log_b y$
For $b > 0$ , $b \neq 1$ , $y > 0$ .
Product law: $\log_b(MN) = \log_b M + \log_b N$
Quotient law: $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$
Power law: $\log_b M^n = n\log_b M$
Special values: $\log_b 1 = 0 \qquad \log_b b = 1 \qquad b^{\log_b x} = x$
Change of base: $\log_b x = \frac{\log x}{\log b}$
Half-life: $A(t) = A_0\left(\frac{1}{2}\right)^{\frac{t}{h}}$
Compound growth: $A = P(1 + i)^n$
pH: $\text{pH} = -\log\big[\text{H}^+\big]$
Earthquake magnitude: $M = \log\left(\frac{I}{I_0}\right) \qquad M_2 - M_1 = \log\left(\frac{I_2}{I_1}\right)$
Sound level (decibels): $\beta = 10\log\left(\frac{I}{I_0}\right) \qquad \beta_2 - \beta_1 = 10\log\left(\frac{I_2}{I_1}\right)$

Combining Functions

Sum and difference: $(f \pm g)(x) = f(x) \pm g(x)$
Domain: where both $f$ and $g$ are defined.
Product: $(f \cdot g)(x) = f(x)\,g(x)$
Quotient: $\left(\frac{f}{g}\right)(x) = \frac{f(x)}{g(x)}$
Also exclude every $x$ where $g(x) = 0$ .
Composition: $(f \circ g)(x) = f\big(g(x)\big)$
$x$ in the domain of $g$ and $g(x)$ in the domain of $f$ .
Inverse check: $(f \circ f^{-1})(x) = x$

MHF4U Formula Sheet FAQ

Do you get a formula sheet on the MHF4U 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 MHF4U formulas should I memorize?+

The log laws, the compound and double angle formulas, the Pythagorean identities and the special angle values. They show up in almost every question in their units, and looking them up every time costs you minutes.

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