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 11As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$ 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 22### Circular loci 23 24$$(x-a)^2 + (y-b)^2 = r^2$$ 25 26point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre) 27 28### Linear loci 29 30$$QP=RP$$ 31$$\sqrt{(x_Q-q_P)^2+(y_Q-y_P)^2} = \sqrt{(x_R-x_P)^2+(y_R-y_P)^2}$$ 32 33points $Q$ and $R$ are fixed and have a perpendicular bisector $QR$. Therefore, any point on line $y=mx+c$ is equidistant from $QP$ and $RP$. 34 35Since the bisector of the line joining points $Q$ and $R$ is perpendicular to $QR$: 36 37$$m(QR) \times m(RP) = -1$$ 38 39### Parabolic loci 40 41$$PD=PF$$ 42$$|y-z|=\sqrt{(x-x_F)^2+(y-y_F)^2}$$ 43$$(y-z)^2=(x-x_F)^2+(y-y_F)^2$$ 44 45Distance of point $P(x,y)$ from fixed point $F(a,b)$ is equal to the distance of $P$ from $y=z \perp$. 46 47Fixed point $F$ is the **focus** (halfway between $y=z$ and $y=y_P$) 48 49Fixed line $x=z$ is the **directrix** 50 51### Elliptical loci 52 53$$F_1 P + F_2 P =k$$ 54 55**Two** foci at $F_1$ and $F_2$