VCE Specialist Mathematics/Units 3 and 4: Specialist Mathematics/Formulae

Preface
This is a list of all formulae needed for Units 3 and 4: Specialist Mathematics.

Ellipses
General formula:


 * $$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$

General Notes:


 * Point $$(h,k)$$ defines the ellipses center.
 * Points $$(\pm a + h,k)$$ defines the ellipses domain, and horizontal endpoints - i.e. horizontal stretch.


 * Points $$(h,\pm b + k)$$ defines the ellipses range, and vertical endpoints - i.e. vertical stretch.

Circles
General formula:


 * $$(x-h)^2 + (y-k)^2 = r^2$$

General Notes:


 * Point $$(h,k)$$ defines the circles center.
 * Points $$(\pm r + h,k)$$ defines the circles domain - i.e. stretch.


 * Points $$(h,\pm r + k)$$ defines the circles range - i.e. stretch.


 * A circle is a subset of an ellipse, such that $$a = b = r$$.

Hyperbolas
General formulae:


 * $$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$$

General Notes:
 * $$\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1$$


 * Point $$(h,k)$$ defines the hyperbolas center.


 * Points $$(\pm a + h,k)$$ defines the hyperbolas domain, $$[\pm a + h, \pm \infty)$$.


 * The switch in positions of the fractions containing x and y, indicate the type of hyperbola - i.e. vertical or horizontal. The hyperbola is horizontal in the first, and negative in the second of the General hyperbolic formulae above.


 * Graphs $$y=\pm (\pm a + h,k)$$ defines the hyperbolas domain $$[\pm a + h, \pm \infty)$$.

Sin
General formula:
 * $$y = a\sin(n(x-b)) + c$$

General Notes:


 * General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".


 * A period is equal to $$[\frac{2\pi}{n}]$$


 * The domain, unless restricted, is $$x \in \mathbb{R}$$


 * The range is equal to $$[\pm a +c]$$, as the range of $$y = \sin(x), y \in [-1,1]$$, see unit circle.


 * The horizontal translation of $$b$$ is reflected in the x-intercepts.

Cos
General formula:


 * $$y = a\cos(n(x-b)) + c$$

General Notes:


 * General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".


 * The domain, unless restricted, is $$x \in \mathbb{R}$$, as $$y = \cos(x), x \in \mathbb{R}$$


 * A period is equal to $$[\frac{2\pi}{n}]$$, as the factor of n


 * The range is equal to $$[\pm a +c]$$, as the range of $$y = \cos(x), y \in [-1,1]$$, see unit circle.


 * The horizontal translation of $$b$$ is reflected in the x-intercepts.

Tan
General formula: General Notes:
 * $$y = a\tan(n(x-b)) + c$$


 * General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".


 * A period is equal to $$[\frac{\pi}{n}]$$


 * The domain, $$x \in \mathbb{R} \setminus \frac{k\pi}{2n}, k \in \mathbb{N} $$, as $$ y = \tan(x), x \in \mathbb{R} \setminus \frac{k\pi}{2}, k \in \mathbb{N}$$, indicating the asymptotes.


 * The range, unless restricted, is $$y \in \mathbb{R}$$, as the range of $$y = \tan(x), y \in \mathbb{R}$$, see unit circle.


 * The horizontal translation of $$b$$ is reflected in the x-intercepts.

Arcsin
Also known as $$Sin^-1$$ or $$sin^-$$

Arccos
Also known as $$Cos^-1$$ or $$cos^-$$

Arctan
Also known as $$Tan^-1$$ or $$tan^-$$