Problem 7 - Intermediate Algebra

Question

Write the standard form of the equation of a hyperbola centered at the origin that opens left and right.

Step-by-Step Solution

Answer

The standard form is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

Step 1: Identify the Standard Form of a Hyperbola Opening Left and Right

The standard form of the equation of a hyperbola that opens left and right is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This is because such a hyperbola opens along the x-axis.

Step 2: Define the Parameters

In this equation, \( a \) represents the distance from the center to each vertex along the x-axis, and \( b \) represents the distance from the center to each co-vertex along the y-axis. The values of \( a \) and \( b \) will determine the shape and size of the hyperbola.

Step 3: Special Case Centered at the Origin

When the hyperbola is centered at the origin, the equation remains the same: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). There are no additional terms needed for translation.

Key Concepts

Standard Form of Hyperbola

When studying hyperbolas, it's crucial to understand what the standard form of the equation looks like. Unlike an ellipse, a hyperbola's equation includes a subtraction in its standard form. For hyperbolas that open left and right, the equation is given as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This specific form tells us that the hyperbola's major axis aligns with the x-axis. Here, \( a \) and \( b \) are key parameters that control the shape and size of the hyperbola.

• \( a \) is the distance from the center to each vertex on the x-axis.
• \( b \) is the distance from the center to each co-vertex on the y-axis.

Understanding these components helps in tweaking the hyperbola equation to match the specific shape and size required. The term \( a^2 \) appears under \( x^2 \) because the transverse axis (the axis the hyperbola opens along) is horizontal.

Centered at the Origin

A hyperbola is considered to be centered at the origin when its center point is at \((0, 0)\). In geometrical terms, the origin acts as the reference point from which the distances \( a \) and \( b \) are measured.

• The origin makes the equation simpler as no shifts along the axes are needed.
• There are no additional terms, like \( h \) or \( k \), in the equation, making it \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).

With the center at the origin, we get the purest form of the hyperbola’s equation, offering a straightforward way to describe its symmetry and behavior about the x and y axes. This simplicity helps in understanding the geometric properties without considering any translations.

Equation of Hyperbola

The equation of a hyperbola captures its defining geometric properties. For a hyperbola that opens horizontally and is centered at the origin, we use the expression \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). This equation highlights how the hyperbola extends infinitely to the left and right. **Key components of the equation:**

• \( x^2 \) under \( a^2 \): Indicates the direction of the opening along the x-axis.
• \( y^2 \) under \( b^2 \): Reflects how the hyperbola stretches away from the x-axis.
• The "-" between the fractions: Essential for distinguishing hyperbolas from other conic sections.

This structure of the equation not just delineates the shape, but also helps in comparing different hyperbolas. Variations in \( a \) and \( b \) alter how wide or narrow the hyperbola appears, giving us control over its exact proportions. Such comprehension aids in graphing and analysis, making it a foundational element for students learning about conic sections.