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

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Mechanics Practice SACS


Preface[edit]

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

Formulae[edit]

Ellipses, Circles and Hyperbolas[edit]

Ellipses[edit]

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[edit]

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[edit]

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[edit]

Sin[edit]

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[edit]

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[edit]

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[edit]

Also known as Sin^-1 or sin^-

Arccos[edit]

Also known as Cos^-1 or cos^-

Arctan[edit]

Also known as Tan^-1 or tan^-