Problem 37 - Algebra 2

Question

Open-Ended Choose two points on an axis to be the vertices of a hyperbola. Choose two other points on the same axis to be the foci. Write the equation of your hyperbola and draw its graph.

Step-by-Step Solution

Answer

The equation of the hyperbola with vertices at (-1, 0) and (1, 0) and foci at (-2, 0) and (2, 0) is \(x^2 - y^2/3 = 1\).

Step 1: Select the Vertices and Foci

Select two points on the x-axis as vertices, say (-1, 0) and (1, 0) and also select two points on the same x-axis as foci, say (-2, 0) and (2, 0).

Step 2: Find the values of parameters a, b, c

The vertices are a units from the center, so here a = 1. The foci are c units from the center, so c = 2. Using the formula for a hyperbola \(b^2 = c^2 - a^2\), then \(b^2 = 2^2 - 1^2 = 3\).

Step 3: Write the Equation

Using the above obtained values of a and b, the equation of the hyperbola on the x-axis becomes \(x^2/1^2 - y^2/√3^2 = 1\), which simplifies to \(x^2 - y^2/3 = 1\).

Step 4: Draw the Graph

The graph of the hyperbola \(x^2 - y^2/3 = 1\) can be drawn by plotting the center, vertices and foci on the x-axis and asymptotes through the center at angles of gradient ±b/a, so gradients ±√3. Two branches of the hyperbola, foating from the vertices, are drawn on the outside of these asymptotes.

Key Concepts

Vertices of a Hyperbola

The vertices of a hyperbola are essential points that help in understanding its structure. They are the points where the hyperbola intersects its transverse axis. In this exercise, we placed the vertices at

• (-1, 0)
• (1, 0)

These vertices are 2 units apart, with the midpoint
(0, 0) (the center of the hyperbola)
right between them. The distance from the center to either vertex is given by the parameter \(a\). For our hyperbola, \(a = 1\) since each vertex is 1 unit away from the center. Knowing the distance to the vertices allows you to set part of the equation for the hyperbola. The parameter \(a\) reflects how "open" the hyperbola is on its axis, forming key components of its equation.

Foci of a Hyperbola

The foci of a hyperbola are crucial in defining its shape. They lie on the same line as the vertices and are always farther from the center. For this exercise, we placed the foci at

• (-2, 0)
• (2, 0)

Each focus is 2 units away from the center, leading to \(c = 2\). The relationship between the foci and the vertices is expressed by the equation \(c^2 = a^2 + b^2\). By using the given vertices and foci, we derived \(b^2 = 3\). The foci are where the distances to any point on the hyperbola remain constant when combined. Additionally, they determine how the hyperbola "stretches" along its axis. This property is what differentiates a hyperbola from an ellipse, marking its distinctive sharp curves.

Graphing Hyperbolas

Graphing a hyperbola is a visually engaging task that involves several key steps. Once the equation is defined—in this case, \(x^2 - \frac{y^2}{3} = 1\)—we can start graphing:

• Identify the center (0, 0), where the axes intersect.
• Plot the vertices at (-1, 0) and (1, 0).
• Mark the foci at (-2, 0) and (2, 0).

A hyperbola's asymptotes are lines that the branches approach but never touch. These form at angles with gradients \(\pm\sqrt{3}\) through the center. Draw these asymptotes to help shape the curve. The hyperbola will form two separate branches on either side, extending out from the vertices and getting closer to but never touching the asymptotes. This sketch gives you a complete picture of the hyperbola's dual structure, as each side mirrors the other across the center.