Problem 70 - Algebra 2

Question

Graph each equation. $$ 4 x^{2}-9 y^{2}=36 $$

Step-by-Step Solution

Answer

The graph of the equation is a hyperbola centered at (0,0), with vertices at (±3,0), foci at (±√13,0), and with asymptotes \(y = ± (2/3)x\).

Step 1: Set the equation in the standard form

The equation should be written in the standard form for a hyperbola. If a hyperbola opens sideways, the equation is \((x - h)^2 / a^2 - (y - k)^2 / b^2 = 1\). For a hyperbola that opens up and down, the equation is \((y - k)^2 / a^2 - (x - h)^2 / b^2 = 1\). The current equation can be rewritten as \((x^2/3^2) - (y^2/2^2) = 1\). So, from the equation, a = 3, b = 2 and the hyperbola is centered at (h, k) = (0, 0).

Step 2: Identify vertices and foci

The vertices of the hyperbola are (±a, 0), or (±3, 0). The foci are at (±c, 0) where \(c = \sqrt{a^2 + b^2} = \sqrt{3^2 + 2^2} = \sqrt{13}\). So the coordinates of the foci are \((± \sqrt{13}, 0)\).

Step 3: Graphing the hyperbola

To graph, start by drawing the center at the origin. Then mark the vertices and foci. Draw the two branches of the hyperbola, making sure they pass through the vertices and approach the asymptotes. The asymptotes are the diagonal lines that pass through the center and establish the path along which the hyperbola approaches as it moves away from the center. For the given equation the asymptotes are \(y = ± (2/3)x\).

Key Concepts

Graphing Hyperbolas

Hyperbolas are fascinating curves shaped somewhat like two mirrored arches. They have two separate parts, known as branches, that veer off to infinity. When graphing hyperbolas, the first step is to identify the center, vertices, and foci. These elements are crucial in shaping and positioning the hyperbola on a graph.

• Start by plotting the center point.
• Then, plot the vertices along the transverse axis.
• The foci, further away from the center than the vertices, guide the curve of each branch.

Use these points to sketch the hyperbola, making sure it hugs closely to the asymptotes. Asymptotes, which will be explained in detail later, are lines that the curve approaches but never quite reaches. With these elements plotted, you can see the hyperbola taking shape on the coordinate plane.

Conic Sections

A hyperbola is one of the four conic sections, the others being the circle, ellipse, and parabola. Conic sections result from slicing a cone at different angles. Hyperbolas occur when the cone is sliced with a plane in such a way that it cuts through both halves of the cone.
This dual-branch feature distinguishes hyperbolas from other conic sections:

• Circles and ellipses are closed curves.
• Parabolas are open curves with one branch.
• Hyperbolas have two distinct branches.

Understanding conic sections provides a broader perspective on hyperbolas and helps in distinguishing characteristics and graphing methods essential for various mathematical applications.

Identifying Vertices and Foci

Vertices and foci are fundamental components of hyperbolas. They help define the shape and orientation of the curve. The vertices are the points where each branch is closest together, while the foci lie outside the vertices, along the transverse axis.
For the equation \( (x^2/3^2) - (y^2/2^2) = 1 \):

• The vertices are at \((±3, 0)\), indicating the points where the branches cross the x-axis.
• The foci, calculated using \(c = \sqrt{a^2 + b^2} = \sqrt{13}\), are at \((±\sqrt{13}, 0)\).

These points are crucial for drawing the hyperbola accurately, as they guide both the size and curvature of the individual branches.

Equation of Hyperbola

The standard form of a hyperbola's equation reveals its orientation and properties. For hyperbolas centered at the origin, the equations are:

• \((x-h)^2/a^2 - (y-k)^2/b^2 = 1\) for hyperbolas opening sideways.
• \((y-k)^2/a^2 - (x-h)^2/b^2 = 1\) for hyperbolas opening upwards and downwards.

Converting the given equation \(4x^2 - 9y^2 = 36\) into the standard form, we rewrite it as \((x^2/3^2) - (y^2/2^2) = 1\). Here:

• The transverse axis is horizontal because \(x^2\) comes first in the equation.
• The center is at the origin \((0,0)\).
By identifying \(a\) and \(b\) from this form, we can plot the vertices and calculate the foci, completing the hyperbola's framework.

Asymptotes of Hyperbola

Asymptotes are lines that the branches of the hyperbola approach but never touch. They play a critical role in graphing because they define the slope and direction of the curve as it extends away from the center. For hyperbolas aligned with the x-axis, the asymptotes form through the origin, following the equation:

• \(y = ±(b/a)x\) for horizontal hyperbolas.
• For our earlier example, this translates to \(y = ±(2/3)x\).

The asymptotes create an "x" shape, and each branch of the hyperbola will curve away on either side, getting closer to the lines without intersecting them. This characteristic helps predict the hyperbola's behavior at infinity and guides the plotting of its branches on the graph.