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# 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 $\quad x \rightarrow  \pm \infty,\quad {-1 \over x}$ -->

\includegraphics[width=0.25\textwidth]{./graphics/recip-parabola.png}
\includegraphics[width=0.25\textwidth]{./graphics/recip-sin-cos.png}

- 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
- graph on CAS - **conics**

### Circular loci

point $P(x,y)$ has a constant distance $r$ from point $C(a,b)$ (centre)


$$PC  = r$$

$$(x-a)^2 + (y-b)^2 = r^2$$



### 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

Point $P$ moves so that the sum of its distances from two fixed points $F_1$ and $F_2$ is a constant $k$.

$${F_1 P}  +   F_2 P  =k$$

**Two** foci at $F_1$ and $F_2$

Cartesian equation for ellipses:
$${(x-h)^2 \over a^2} + {(y-k)^2 \over b^2} = 1$$
centered at $(h,k)$. Width is $2a$, height is $2b$.

### Transformations
$$(x,y) \rightarrow (x \prime, y \prime)$$

where $x \prime$ and $y \prime$ are the transformation factors (dilation away from $x$-axis means coefficient of $y$ increases in $y \prime$, and vice versa).

Transformed equation is the same as initial equation with each term divided by its dilation coefficients (must be in terms of $x\prime$ and $y\prime$).

e.g.

$x^2 + y^2 = 1$ is dilated $3$ from $x$, $5$ from $y$.
Transformation rule is $(x\prime,y\prime) = (5x,3y)$
$x={x\prime \over 5},\quad y={y\prime \over 3}$

Equation $x^2 + y^2=1$ becomes

$${(x\prime)^2 \over 25}+ {(y\prime)^2 \over 9}=1$$


### Hyperbolic loci

$$|(F_2P - F_1P  )| = k$$

Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$

Distance between vertices is $2a$
Vertices given by $(h \pm a, k)$

Asymptotes at $y=\pm {b \over a}(x-h)+k$
To make hyperbola up/down rather than left/right, swap $x$ and $y$

$y^2-x^2=1$ produces hyperbola shifted 90 $^\circ$ (top and bottom of asymptotes)

## Parametric equations

Parametric curve:

$$x=f(t), \quad y=g(t)$$

$t$ is the parameter

To convert to cartesian, solve like simultaneous equations

## Polar coordinates

$$x = r\cos\theta, \quad y = r\sin\theta$$

### Spirals
$$r={\theta \over n\pi}$$
- solve intercepts for multiples of $\pi \over 2$
- or draw table of values for $r$ and $\theta$ for each $n\pi \over 2$

### Circles
$$r=a$$

### Lines

Horizontal: $r={n \over \sin \theta}$
Vertical: $r={n \over \cos \theta}$

### Cardioids

$$r=a(n+ \cos\theta)$$

### Roses

$$r=\cos(k\theta)$$

If $k$ is odd, half of the petals will overlap (hence there are $n$ petals)

If $k$ is even, petals will not overlap (hence $2n$ petals)

\includegraphics[width=0.5\textwidth]{./graphics/rose.png}


### Solving polar graphs

solve in terms of $r$

e.g. $x=4$

$r\cos\theta = 4$

$r={4 \over \cos\theta}$


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e.g. $y=x^2$

$r\sin\theta = r^2 \cos^2 \theta$

$\sin \theta = r \cos^2 \theta$

$r = {\sin \theta \over \cos^2\theta} = \tan\theta \sec\theta$

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e.g. $r=6\cos \theta\quad$ *(multiply by $r$)*

$r^2=6r\cos\theta$

$x^2+y^2=6x$

complete the square