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.

Formulae

Ellipses, Circles and Hyperbolas

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:

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

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

General Notes:

• 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 $[\pm a+h,\pm \infty )$.

Trignometric Functions

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:

• $y=a\tan(n(x-b))+c$

General Notes:

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

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

• 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 $tan^{-}$