Question

For the following exercises, sketch the graph of each conic. $$ \frac{x^{2}}{16}-\frac{y^{2}}{9}=1 $$

Short answer

The hyperbola is centered at the origin with vertices \((4,0)\) and \((-4,0)\), and asymptotes \(y = \pm \frac{3}{4}x\).

Step-by-step solution

Identify the Type of Conic

The given equation \( \frac{x^{2}}{16} - \frac{y^{2}}{9} = 1 \) is a standard form of a hyperbola. This is identified by the fact that it has the difference of squares on the left-hand side with a positive term (\(x^2/16\)) and a negative term (\(y^2/9\)), and it equals 1.

Determine Orientation

Since the \(x^2\) term is positive and comes first in the equation, the hyperbola is oriented horizontally. This means the hyperbola will open to the left and right.

Identify the Center

The center of the hyperbola is at the origin \((0,0)\), as there are no \(h\) or \(k\) values subtracted inside \(x\) or \(y\) terms.

Calculate Vertices

The vertices are calculated from the \(\frac{x^{2}}{16}\) term. \(a^2 = 16\) implies \(a = 4\). The vertices are located at \((\pm a, 0) = (4, 0)\) and \((-4, 0)\).

Calculate Asymptotes

Using the formula for the asymptotes of a horizontal hyperbola: \(y = \pm \frac{b}{a}x\). Here, \(b^2 = 9\) so \(b = 3\). Therefore, the asymptotes are \(y = \pm \frac{3}{4}x\).

Sketch the Graph

Plot the center at \((0,0)\) on a graph, mark the vertices at \((4,0)\) and \((-4,0)\), and draw the asymptotes \(y = \frac{3}{4}x\) and \(y = -\frac{3}{4}x\). Draw the hyperbola branches opening horizontally passing through the vertices and approaching the asymptotes without touching them.

Key concepts

Equation of Hyperbola

A hyperbola is a type of conic section represented by an equation with a distinctive form. The general equation is: \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] This is the standard form for a hyperbola that opens horizontally. It features a subtraction between two squared terms, indicating the hyperbola's shape. Key components include:

• Center: Point \(h, k\) where the hyperbola is centered. Here, both are zero as there's no modifications within the terms.
• Terms: \(a^2\) and \(b^2\) determine the size and orientation.
• Orientation: The term before the subtraction tells us how the hyperbola opens (either along the x-axis or y-axis).

Understanding this standard equation enables quick identification of a hyperbola's key features in any problem.

Conic Sections

Conic sections come from the intersection of a plane with a cone, creating shapes like ellipses, parabolas, and hyperbolas. Each type has unique properties and equations. Hyperbolas are particularly distinct as they consist of two separate curves.

• Types of Conic Sections: Circle, Ellipse, Parabola, Hyperbola.
• Hyperbolas: Are characterized by having two curves that mirror each other, reflecting complex interaction derived from their geometric plane intersection.
• Applications: Found in orbits, physics involving fields, and architectural structures.

The unique geometric result that produces hyperbolas makes understanding their properties valuable in multiple mathematical and practical applications.

Graphing Hyperbolas

Graphing a hyperbola involves plotting points to capture its unique double curve shape accurately. Start by identifying the hyperbola's essential components from its equation:

• Center: Location of the hyperbola's midpoint, before the curves diverge.
• Vertices: Major points along the hyperbola that determine its width. Calculated using \(a \), the distance from the center.
• Orientation: Direction that the hyperbola opens, which can be along the x or y-axis.

Draw the asymptotes first, which guide the hyperbola's shape but aren't part of the hyperbola. Finally, sketch the curves that approach, but never touch these asymptote lines, ensuring the distinctive open-mouthed structure is portrayed.

Asymptotes of Hyperbolas

Asymptotes are straight lines a hyperbola approaches but never reaches. They provide a framework or boundary within which the hyperbola exists. For hyperbolas, asymptotes assist in defining the overall shape.

• Equation for Asymptotes: Derived from the hyperbola's orientation and coefficients, for example: \ y = \pm \frac{b}{a}x \ for horizontal hyperbolas.
• Purpose: Serve as guides for sketching hyperbolas, illustrating curve behavior as it extends infinitely far away from the center.
• Calculation: Involves the terms \(a\) and \(b\) from the hyperbola's equation, dictating the steepness of the asymptote lines.

In summary, asymptotes are crucial for understanding how a hyperbola extends into space, making them fundamental in graphing and visualizing the conic section effectively.