11.4 Hyperbolas

Hyperbolas are fascinating conic sections with unique shapes and properties. They're defined by equations with a minus sign between terms, creating two separate branches that extend infinitely. Unlike circles or ellipses, hyperbolas have asymptotes they approach but never touch.

Key features of hyperbolas include vertices, foci, and transverse axes. Their equations can be centered at the origin or any point (h,k). Understanding how to graph hyperbolas and identify their characteristics is crucial for mastering this topic in algebra.

Hyperbolas

Graphing hyperbolas and key features

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• Hyperbolas centered at the origin
  • Transverse axis along the x-axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$
    • Vertices located at $(\pm a, 0)$ represent the points where the hyperbola intersects the transverse axis
    • Foci located at $(\pm c, 0)$, where $c^2 = a^2 + b^2$, are two fixed points that define the shape and symmetry of the hyperbola
    • Asymptotes with equations $y = \pm \frac{b}{a}x$ are diagonal lines that the hyperbola approaches but never touches as it extends to infinity (rectangular hyperbola)
  • Transverse axis along the y-axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$
    • Vertices located at $(0, \pm a)$ represent the points where the hyperbola intersects the transverse axis
    • Foci located at $(0, \pm c)$, where $c^2 = a^2 + b^2$, are two fixed points that define the shape and symmetry of the hyperbola
    • Asymptotes with equations $y = \pm \frac{a}{b}x$ are diagonal lines that the hyperbola approaches but never touches as it extends to infinity
• Hyperbolas centered at (h,k)
  • Transverse axis parallel to the x-axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
    • Vertices located at $(h \pm a, k)$ represent the points where the hyperbola intersects the transverse axis
    • Foci located at $(h \pm c, k)$, where $c^2 = a^2 + b^2$, are two fixed points that define the shape and symmetry of the hyperbola
    • Asymptotes with equations $y - k = \pm \frac{b}{a}(x - h)$ are diagonal lines that the hyperbola approaches but never touches as it extends to infinity
  • Transverse axis parallel to the y-axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$
    • Vertices located at $(h, k \pm a)$ represent the points where the hyperbola intersects the transverse axis
    • Foci located at $(h, k \pm c)$, where $c^2 = a^2 + b^2$, are two fixed points that define the shape and symmetry of the hyperbola
    • Asymptotes with equations $y - k = \pm \frac{a}{b}(x - h)$ are diagonal lines that the hyperbola approaches but never touches as it extends to infinity

Equations of hyperbolas

• Standard form equations
  • Transverse axis along the x-axis: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ represents a hyperbola centered at the origin with vertices on the x-axis
  • Transverse axis along the y-axis: $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ represents a hyperbola centered at the origin with vertices on the y-axis
• General form equations
  • Transverse axis parallel to the x-axis: $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ represents a hyperbola centered at (h,k) with vertices on a horizontal line
  • Transverse axis parallel to the y-axis: $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ represents a hyperbola centered at (h,k) with vertices on a vertical line
• Determining equations from graphs or characteristics
  1. Identify the center (h,k) by locating the point of symmetry and determine the direction of the transverse axis (horizontal or vertical)
  2. Determine the vertices by finding the points where the hyperbola intersects the transverse axis and calculate the length of the transverse axis (2a)
  3. Find the length of the conjugate axis (2b) using the slope of the asymptotes, which can be determined by the angle they make with the transverse axis
  4. Substitute the values of h, k, a, and b into the appropriate general form equation based on the direction of the transverse axis

Hyperbolas vs other conic sections

• Hyperbolas
  • Equations in the form $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ with a minus sign between the terms
  • Graphs consist of two separate branches that extend infinitely and approach two asymptotes (diagonal lines)
  • The eccentricity of a hyperbola is always greater than 1, which distinguishes it from other conic sections
• Parabolas
  • Equations in the form $(y-k)^2 = 4p(x-h)$ or $(x-h)^2 = 4p(y-k)$ with a single squared term and a linear term
  • Graphs have a single U-shaped curve with a vertex (turning point) and a directrix (line) that defines its shape
• Ellipses
  • Equations in the form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ with a plus sign between the terms and squared variables in both the numerator and denominator
  • Graphs are closed curves with two axes of symmetry (major and minor) and two foci that define their shape
• Circles
  • Equations in the form $(x-h)^2 + (y-k)^2 = r^2$ with squared terms for both variables and a constant term on the right side
  • Graphs are closed curves with a center (h,k) and a radius r, where all points on the circle are equidistant from the center (special case of an ellipse with equal axes)

Additional Hyperbola Properties

• Latus rectum: The line segment perpendicular to the transverse axis through a focus, with endpoints on the hyperbola
• Directrix: A line perpendicular to the transverse axis that helps define the shape of the hyperbola in relation to its focus
• Degenerate hyperbola: A special case where the hyperbola reduces to two intersecting lines, occurring when the equation's discriminant equals zero
• Hyperbolic functions: Mathematical functions (sinh, cosh, tanh) that are closely related to the geometry of hyperbolas, similar to how trigonometric functions relate to circles