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, $[\pm a+h,\pm \mathrm{\infty })$.

• 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 $[\pm a+h,\pm \mathrm{\infty })$.

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 $x\in ℝ$

• 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 $x\in ℝ$, as $y=\cos (x),x\in ℝ$

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

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

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

Arcsin

Also known as $Si{n}^{-}1$ or $si{n}^{-}$

Arccos

Also known as $Co{s}^{-}1$ or $co{s}^{-}$

Arctan

Also known as $Ta{n}^{-}1$ or $ta{n}^{-}$