From 7e58f46f46f9e945a5ecb60534f166f8fad2082a Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Fri, 22 Jun 2018 10:42:36 +1000 Subject: [PATCH] start spec graphing techniques --- spec/graphing.md | 55 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 55 insertions(+) create mode 100644 spec/graphing.md diff --git a/spec/graphing.md b/spec/graphing.md new file mode 100644 index 0000000..94c5a4a --- /dev/null +++ b/spec/graphing.md @@ -0,0 +1,55 @@ +# Graphing techniques + +## Reciprocal continuous functions + +If $y=f(x)$, the reciprocal function is: + +$$y={1 \over f(x)}$$ + +As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (vert asymptote at $f(x)=0$) + +As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$ + +- reciprocal functions are always on the same side of $x=0$ +- if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$ +- point of inflection at $P(1,1)$ + +## Locus of points + +- set of points that satisfy a given condition +- path traced by a point that moves according to a condition + +### Circular loci + +$$(x-a)^2 + (y-b)^2 = r^2$$ + +point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre) + +### Linear loci + +$$QP=RP$$ +$$\sqrt{(x_Q-q_P)^2+(y_Q-y_P)^2} = \sqrt{(x_R-x_P)^2+(y_R-y_P)^2}$$ + +points $Q$ and $R$ are fixed and have a perpendicular bisector $QR$. Therefore, any point on line $y=mx+c$ is equidistant from $QP$ and $RP$. + +Since the bisector of the line joining points $Q$ and $R$ is perpendicular to $QR$: + +$$m(QR) \times m(RP) = -1$$ + +### Parabolic loci + +$$PD=PF$$ +$$|y-z|=\sqrt{(x-x_F)^2+(y-y_F)^2}$$ +$$(y-z)^2=(x-x_F)^2+(y-y_F)^2$$ + +Distance of point $P(x,y)$ from fixed point $F(a,b)$ is equal to the distance of $P$ from $y=z \perp$. + +Fixed point $F$ is the **focus** (halfway between $y=z$ and $y=y_P$) + +Fixed line $x=z$ is the **directrix** + +### Elliptical loci + +$$F_1 P + F_2 P =k$$ + +**Two** foci at $F_1$ and $F_2$ -- 2.43.2