Rational function graphing

$y=\frac{b{x}^m+...}{b{x}^n+...}$.

when there's a hole, steps

• The hole x value, x<>
• Vertical asymptotes (when denominator is equal to 0)
• Horizontal asymptotes (degree m of numerator vs degree n of the denominator)
  • if m < n, y=0
  • if m = n, y=b/a
  • if m > n, no horizontal asymptote does not exist
• y-int and x-int
• hole coordinates
• pick values on each side of vertical asymptotes for graphing

Conic sections of a double-napped cone

• general form of the equation: $A{x}^2+C{y}^2+Dx+Ey+F=0$
  • Circle: A=C<>0
  • Ellipse: AC>0 and A<>C
  • Hyperbola: AC<0
  • Horizontal parabola: A=0, CD<>0 (C<>0 and D<>0)
  • Vertical parabola: C=0, AE<>0 (A<>0 and E<>0)

Circle

• the set of all points (x, y) in a plane that are equidistant from a fixed point (centre).

Ellipse

• the set of all points (x, y) in a plane such that the sum of the distances between any point on the ellipse and 2 fixed points (foci) is a constant
  • the major and minor axes
    • if a>b, 2a, 2b
    • if a<b, 2b, 2a
  • if a>b, $c=\pm \sqrt{a^2-b^2}$.

Parabola

the set of all points (x, y) in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix)

• will eat the focus
• vertex (h, k)
  • $(y-k{)}^2=4p(x-h)$ - focus (p+h, k), directrix x=-p+h
  • $(x-h{)}^2=4p(y-k)$ - focus (h, p+k), directrix y=-p+k
• distance b/w focus and the directrix focus-directrix = 2p
• distance b/w vertex and directrix = p
• distance b/w focus and vertex = p
• watch out for the order of coordinates when translating from the equation
• open to left and right $y^2=4px$,
• open to up and down $x^2=4py$
• Hyperbola - the set of all points (x, y) in a plane such that the difference of the distances b/w any point on the hyperbola and 2 fixed points (foci) is a constant

Test

• Writing an equation of a conic section given "clues" or a graph
• Identify conic sections