Hyperbola Conic Section: Equations, Graphs, and Translations

Geometric Formation of a Hyperbola Conic Section

• A hyperbola is formed as a conic section when a plane intersects both parts of a double knot cone (also referred to as a double napped cone).
• This intersection creates two distinct curves, which the tutorial identifies as two parabolas.
• The distance between the vertexes of these parabolas varies based on the relative position of the intersecting plane:     * As the plane moves toward the outside of the double napped cone, the distance between the vertexes of the parabolas increases.     * As the plane moves toward the center of the double napped cone, the distance between the vertexes of the parabolas decreases.

Standard Equation of a Hyperbola Centered at the Origin

• The general equation for a hyperbola centered at the origin $(0, 0)$ is expressed as:     * $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
• In this equation, the values for the constants $a$ and $b$ will always be provided for the student.
• This specific form indicates that the hyperbola is centered at the coordinates $(0, 0)$.

General Equation and Translations (h, k)

• Hyperbolas can be translated from the origin to a new center using horizontal and vertical shifts.
• The translated form of the equation is:     * $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$
• The variables $h$ and $k$ represent the following:     * $h$: This represents the horizontal translation of the hyperbola.     * $k$: This represents the vertical translation of the hyperbola.
• Under this form, the hyperbola is considered to be centered at the point $(h, k)$.

Practical Example of Hyperbola Translation

• Consider a hyperbola originally centered at the origin $(0, 0)$.
• If the hyperbola is shifted $2$ units to the left and $3$ units up, the equation and graph change accordingly:     * To move the hyperbola $2$ units to the left, the value of $h$ is replaced with $-2$.     * To move the hyperbola $3$ units up, the value of $k$ is replaced with $3$.
• The resulting hyperbola is now centered at the coordinate point where $x = -2$ and $y = 3$.

Summary of Key Concepts

• The general equation for a hyperbola follows the structure involving $x^2$, $y^2$, $a^2$, and $b^2$, set equal to $1$.
• The parameters $a$ and $b$ are given constants.
• The center of the hyperbola is defined by the point $(h, k)$.