Graphing Hyperbolas

Key Questions

What information do you need to graph hyperbolas?
If It's known the equation of the hyperbolas, that is:

$\frac{{(x - x_{c})}_{c}^{2}}{a^{2}} - \frac{{(y - y_{c})}_{c}^{2}}{b^{2}} = \pm 1$ $(x-x_c)^2/a^2-(y-y_c)^2/b^2=+-1$ ,

we can graph the hyperbolas in this way:

find the center $C (x_{c}, y_{c})$ $C(x_c,y_c)$ ;

make a rectangle with the center in $C$ $C$ and with sides $2 a$ $2a$ and $2 b$ $2b$ ;

draw the lines that pass from the opposite vertices of the rectangle (the asymptotes);

if the sign of $1$ $1$ is $+$ $+$ , than the two branches are left and right of the rectangule and the vertices are in the middle of the vertical sides, if the sign of $1$ $1$ is $-$ $-$ , than the two branches are up and down of the rectangule and the vertices are in the middle of the horizontal sides.
Massimiliano · 1 · Mar 13 2015
How do I find an equation for a hyperbola, given its graph?
Answer:

check the explanation below.
Explanation:
$A x^{2} + B y^{2} + C x y + D x + E y + F = 0$ $Ax^2+By^2+Cxy+Dx+Ey+F=0$

That's the general equation of any conic section including the hyperbola

where you can find the equation of a hyperbola given enough points

where $A, B, C, D, E, F$ $color(green)(" A,B,C,D,E,F$ are constants
$where either A or B are negative but never both$ $color(red)("where either A or B are negative but never both"$

though this is usually too hard to solve this type of equation manually

$Special Case:$ $color(blue)("Special Case:"$
the hyperbola is horizontal or vertical

if it's horizontal you could use this

$\frac{{(x - h)}^{2}}{a^{2}} - \frac{{(y - k)}^{2}}{a^{2} (e^{2} - 1)} = 1$ $color(green)((x-h)^2/a^2-(y-k)^2/(a^2(e^2-1))=1$
$where a is a constant and e is the eccentricity (h, k) is the center of the hyperbola$ $color(green)("where a is a constant and "e" is the eccentricity "(h,k)" is the center of the hyperbola"$

$Example :$ $color(blue)("Example :")$

graph{(x-1)^2-(y+2)^2/3=1 [-3.502, 5.262, -3.852, 0.533]}
given one of it's foci points is $(3, - 2)$ $(3,-2)$

Using the properties of the hyperbola to determine the constants

the distance between the two vertices equals $2 a$ $2a$
$2 a = 2$ $2a=2$ $\to$ $rarr$ $a = 1$ $a=1$

now to get the center $(h, k)$ $(h,k)$

it can simply be done by finding the midpoint of the line segment joining the two vertices

$(\frac{0 + 2}{2}, \frac{- 2 - 2}{2}) = (1, - 2)$ $((0+2)/2,(-2-2)/2)=(1,-2)$

now finally to find $e$ $e$

$a e$ $ae$ is the distance between the center and the focus

$a e = 3 - 1 = 2$ $ae=3-1=2$

$e = 2$ $e=2$

$Finally by substituting$ $color(blue)("Finally by substituting"$

you get that the hyperbola's equation is as follows

$\frac{{(x - 1)}^{2}}{1} - \frac{{(y + 2)}^{2}}{2} = 1$ $(x-1)^2/1-(y+2)^2/2=1$

$Additional tricks that could help :)$ $color(green)("Additional tricks that could help :)"$

distance between the directrix and the center of the hyperbola $\frac{a}{e}$ $color(green)(a/e$

if it's vertical the equation will change but with the same solving technique and it will be

$\frac{{(y - k)}^{2}}{a^{2}} - \frac{{(x - h)}^{2}}{a^{2} (e^{2} - 1)} = 1$ $color(green)((y-k)^2/a^2-(x-h)^2/(a^2(e^2-1))=1$
Yahia M. · 1 · Jun 16 2018

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Geometry of a Hyperbola

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Graphing Hyperbolas

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Geometry of a Hyperbola