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Chapter 10: Problem 4

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

Short Answer

Expert verified
The graphic description of the equation shows that it is a hyperbola centered at the origin with vertices at points (2,0) and (-2,0), and asymptotes through the origin at slopes of 卤1.

Step by step solution

01

Identify the Values of \(a\) and \(b\)

The given equation can be rewritten as \(x^2/4 - y^2/1 = 1\). This allows us to identify \(a^2 = 4\) and \(b^2 = 1\), thus \(a = 2\) and \(b = 1\).
02

Determine the Vertices and Foci

The vertices are found at \((\pm a,0)\) and the foci are found at \((\pm \sqrt{a^2+b^2}, 0)\). For our problem, the vertices are at points (2,0) and (-2,0), and the foci are at points \((\pm \sqrt{4+1},0)\) or (卤2.24, 0).
03

Graph the Equation

Sketch the x-axis and y-axis. Mark the center at the origin (0,0), the vertices (卤2, 0), and the foci (卤2.24 ,0). Also, plot the asymptote lines with slopes of 卤1 passing through the origin. These serve as boundary lines of the hyperbola. Now you can outline your hyperbola. The branches of the hyperbola pass through the vertices, wrapping outwards near the boundary lines. Then sketch the final graph of the equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conic Sections
Conic sections are curves obtained by intersecting a plane with a cone. The four basic types are circles, ellipses, parabolas, and hyperbolas. In this exercise, we focus on the hyperbola. Hyperbolas have two separate curves, known as branches. The plane cuts through both halves of the cone, creating this unique shape. These curves are symmetrical and face away from each other.
  • Each conic section has a standard form equation, specific to its type.
  • Hyperbolas are written in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \).
  • They have distinctive properties compared to other conics, notably having asymptotes.
Understanding conic sections helps in various fields like physics, engineering, and computer graphics.
Graphing Equations
Graphing equations of hyperbolas involves rewriting the equation into a recognizable form. In the original exercise, the equation was \( x^2 - 4y^2 = 4 \), and we transformed it into \( \frac{x^2}{4} - \frac{y^2}{1} = 1 \).
  • Identify \(a^2\ = 4\) and \( b^2 = 1 \). Thus, \( a = 2 \) and \( b = 1 \).
  • The equation is now prepared for graphing, showing it as a hyperbola.
Graphing requires placing key points like vertices and asymptotes on the coordinate plane. Hyperbolas open along real number axis determined by \(a\).
This setup ensures clarity in how the graph should be drawn and interpreted.
Vertices and Foci
Vertices and foci are critical in defining the shape and position of a hyperbola. Vertices are points where the hyperbola intersects its transverse axis.
  • From the values \( a = 2 \) and \( b = 1 \), vertices are \( (\pm 2,0) \).
  • Foci are calculated using \( \sqrt{a^2 + b^2} \), yielding \( (\pm 2.24, 0) \).
These points give hyperbolas their distinctive shape. Vertices indicate the "closest" points of the branches, while foci illustrate the core point around which the hyperbola stretches. Marking these on a graph helps visualize the symmetry and spread.
Asymptotes
Asymptotes are invisible lines that the branches of a hyperbola approach but never touch. They guide the general direction of the hyperbolic curves.
  • For our equation, the slopes of the asymptotes are \( \pm \frac{b}{a} = \pm 1 \).
  • These lines pass through the center of the hyperbola, which is at the origin (0,0).
By sketching asymptotes as diagonal lines, we define the open boundaries of the curve. Asymptotes help in predicting how the branches will extend without ever intersecting or overlapping each other. This concept helps in ensuring the graph accurately represents the function.

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