spec / graphing.mdon commit Merge branch 'master' of ssh://charles/tank/andrew/school/notes (5e2fc0c)
   1# Graphing techniques
   2
   3## Reciprocal continuous functions
   4
   5If $y=f(x)$, the reciprocal function is:
   6
   7$$y={1 \over f(x)}$$
   8
   9As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (vert asymptote at $f(x)=0$)
  10
  11<!-- As $\quad x \rightarrow  \pm \infty,\quad {-1 \over x}$ -->
  12
  13- reciprocal functions are always on the same side of $x=0$
  14- if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$
  15- point of inflection at $P(1,1)$
  16
  17## Locus of points
  18
  19- set of points that satisfy a given condition
  20- path traced by a point that moves according to a condition
  21- graph on CAS - **conics**
  22
  23### Circular loci
  24
  25point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre)
  26
  27
  28$$PC  = r$$
  29
  30$$(x-a)^2 + (y-b)^2 = r^2$$
  31
  32
  33
  34### Linear loci
  35
  36$$QP  =  RP $$
  37$$\sqrt{(x_Q-q_P)^2+(y_Q-y_P)^2} = \sqrt{(x_R-x_P)^2+(y_R-y_P)^2}$$
  38
  39points $Q$ and $R$ are fixed and have a perpendicular bisector $QR$. Therefore, any point on line $y=mx+c$ is equidistant from $QP$ and $RP$.
  40
  41Since the bisector of the line joining points $Q$ and $R$ is perpendicular to $QR $:
  42
  43$$m( QR ) \times m( RP ) = -1$$
  44
  45### Parabolic loci
  46
  47$$PD  =  PF $$
  48$$|y-z|=\sqrt{(x-x_F)^2+(y-y_F)^2}$$
  49$$(y-z)^2=(x-x_F)^2+(y-y_F)^2$$
  50
  51Distance of point $P(x,y)$ from fixed point $F(a,b)$ is equal to the distance of $P$ from $y=z \perp$.
  52
  53Fixed point $F$ is the **focus** (halfway between $y=z$ and $y=y_P$)
  54
  55Fixed line $x=z$ is the **directrix**
  56
  57### Elliptical loci
  58
  59Point $P$ moves so that the sum of its distances from two fixed points $F_1$ and $F_2$ is a constant $k$.
  60
  61$${F_1 P}  +   F_2 P  =k$$
  62
  63**Two** foci at $F_1$ and $F_2$
  64
  65Cartesian equation for ellipses:
  66$${(x-h)^2 \over a^2} + {(y-k)^2 \over b^2} = 1$$
  67centered at $(h,k)$. Width is $2a$, height is $2b$.
  68
  69### Transformations
  70$$(x,y) \rightarrow (x \prime, y \prime)$$
  71
  72where $x \prime$ and $y \prime$ are the transformation factors (dilation away from $x$-axis means coefficient of $y$ increases in $y \prime$, and vice versa).
  73
  74Transformed equation is the same as initial equation with each term divided by its dilation coefficients (must be in terms of $x\prime$ and $y\prime$).
  75
  76e.g.
  77
  78$x^2 + y^2 = 1$ is dilated $3$ from $x$, $5$ from $y$.
  79Transformation rule is $(x\prime,y\prime) = (5x,3y)$
  80$x={x\prime \over 5},\quad y={y\prime \over 3}$
  81
  82Equation $x^2 + y^2=1$ becomes
  83
  84$${(x\prime)^2 \over 25}+ {(y\prime)^2 \over 9}=1$$
  85
  86
  87### Hyperbolic loci
  88
  89$$|(F_2P - F_1P  )| = k$$
  90
  91Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
  92$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
  93
  94Asymptotes at $y=\pm {b \over a}(x-h)+k$
  95To make hyperbola up/down rather than left/right, swap $x$ and $y$
  96
  97## Parametric equations
  98
  99Parametric curve:
 100
 101$$x=f(t), \quad y=g(t)$$
 102
 103$t$ is the parameter