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

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Preface

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

Formulae

Ellipses, Circles and Hyperbolas

Ellipses

General formula:

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

General Notes:

• Point ${\displaystyle (h,k)}$ defines the ellipses center.
• Points ${\displaystyle (\pm a+h,k)}$ defines the ellipses domain, and horizontal endpoints - i.e. horizontal stretch.
• Points ${\displaystyle (h,\pm b+k)}$ defines the ellipses range, and vertical endpoints - i.e. vertical stretch.

Circles

General formula:

• ${\displaystyle (x-h)^{2}+(y-k)^{2}=r^{2}}$

General Notes:

• Point ${\displaystyle (h,k)}$ defines the circles center.
• Points ${\displaystyle (\pm r+h,k)}$ defines the circles domain - i.e. stretch.
• Points ${\displaystyle (h,\pm r+k)}$ defines the circles range - i.e. stretch.
• A circle is a subset of an ellipse, such that ${\displaystyle a=b=r}$.

Hyperbolas

General formulae:

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

General Notes:

• Point ${\displaystyle (h,k)}$ defines the hyperbolas center.
• Points ${\displaystyle (\pm a+h,k)}$ defines the hyperbolas domain, ${\displaystyle [\pm a+h,\pm \infty )}$.
• 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 ${\displaystyle y=\pm (\pm a+h,k)}$ defines the hyperbolas domain ${\displaystyle [\pm a+h,\pm \infty )}$.

Trignometric Functions

Sin

General formula:

• ${\displaystyle y=a\sin(n(x-b))+c}$

General Notes:

• General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".
• A period is equal to ${\displaystyle [{\frac {2\pi }{n}}]}$
• The domain, unless restricted, is ${\displaystyle x\in \mathbb {R} }$
• The range is equal to ${\displaystyle [\pm a+c]}$, as the range of ${\displaystyle y=\sin(x),y\in [-1,1]}$, see unit circle.
• The horizontal translation of ${\displaystyle b}$ is reflected in the x-intercepts.

Cos

General formula:

• ${\displaystyle y=a\cos(n(x-b))+c}$

General Notes:

• General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".
• The domain, unless restricted, is ${\displaystyle x\in \mathbb {R} }$, as ${\displaystyle y=\cos(x),x\in \mathbb {R} }$
• A period is equal to ${\displaystyle [{\frac {2\pi }{n}}]}$, as the factor of n
• The range is equal to ${\displaystyle [\pm a+c]}$, as the range of ${\displaystyle y=\cos(x),y\in [-1,1]}$, see unit circle.
• The horizontal translation of ${\displaystyle b}$ is reflected in the x-intercepts.

Tan

General formula:

• ${\displaystyle y=a\tan(n(x-b))+c}$

General Notes:

• General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".
• A period is equal to ${\displaystyle [{\frac {\pi }{n}}]}$
• The domain, ${\displaystyle x\in \mathbb {R} \setminus {\frac {k\pi }{2n}},k\in \mathbb {N} }$, as ${\displaystyle y=\tan(x),x\in \mathbb {R} \setminus {\frac {k\pi }{2}},k\in \mathbb {N} }$, indicating the asymptotes.
• The range, unless restricted, is ${\displaystyle y\in \mathbb {R} }$, as the range of ${\displaystyle y=\tan(x),y\in \mathbb {R} }$, see unit circle.
• The horizontal translation of ${\displaystyle b}$ is reflected in the x-intercepts.

Arcsin

Also known as ${\displaystyle Sin^{-}1}$ or ${\displaystyle sin^{-}}$

Arccos

Also known as ${\displaystyle Cos^{-}1}$ or ${\displaystyle cos^{-}}$

Arctan

Also known as ${\displaystyle Tan^{-}1}$ or ${\displaystyle tan^{-}}$