Problem 62 - Precalculus: Mathematics for Calculus

Question

Use a graphing device to graph the conic. $$x^{2}-4 y^{2}+4 x+8 y=0$$

Step-by-Step Solution

Answer

The conic is a horizontally opening hyperbola centered at \((-2, 1)\).

Step 1: Rearrange the Equation

The given equation is \(x^{2}-4y^{2}+4x+8y=0\). The first step in graphing a conic is to rearrange the equation into a standard conic form. Begin by grouping the \(x\) and \(y\) terms:\((x^2 + 4x) - (4y^2 - 8y) = 0\).

Step 2: Complete the Square for Each Group

We now complete the square for both the \(x\) and \(y\) groups. For \(x\):\(x^2 + 4x\) becomes \((x+2)^2 - 4\).For \(y\):\(-4(y^2 - 2y)\) becomes \(-4((y-1)^2 - 1) = -4(y-1)^2 + 4\).Substitute these back into the equation: \((x+2)^2 - 4 - 4(y-1)^2 + 4 = 0\).

Step 3: Simplify and Identify Conic Type

After simplification, the equation becomes:\((x+2)^2 - 4(y-1)^2 = 0\).Reorganize to:\((x+2)^2 = 4(y-1)^2\).This equation can be recognized as the standard form of a hyperbola because it represents two squares set equal to one another with opposite signs on their squares.

Step 4: Graph the Hyperbola

To graph the hyperbola \((x+2)^2 = 4(y-1)^2\), note the center at \((-2, 1)\). The hyperbola opens horizontally because the \(x\) term is on one side of the equation and the \(y\) term is on the other. The vertices are placed on the line \(y = 1\) at distances determined by the equation.Using a graphing device, plot the center at \((-2,1)\) and plot the vertices by moving a distance corresponding to the relation \((x+2) = ext{constant} \) for the hyperbola following the transformation \((x+2) = ext{positive or negative value of } 2(y-1)\).

Key Concepts

Hyperbola

A hyperbola is a type of conic section, which is formed by slicing through a cone with a plane. This results in two distinct curves that look like an "open" pair of mirrored arcs. Hyperbolas have an important property: they exhibit symmetry across both axes. Hyperbolas are defined by the difference of distances from each point on the curve to the two fixed points known as the foci. This property means that hyperbolas have wider applications from navigation systems to astronomy.

In mathematical terms, hyperbolas are represented by the general form of the equation:

• \( Ax^2 - By^2 + Cx + Dy + E = 0 \)

To recognize this equation as that of a hyperbola, note the difference in sign between the squared terms. This distinction is crucial since it indicates the curve's unique open shape as opposed to other conic sections like ellipses and circles.

Completing the Square

Completing the square is a technique used to simplify quadratic expressions and equations. This mathematical process allows us to transform equations into a form that's easier to interpret, particularly when graphing conics. It essentially rearranges a quadratic equation into a perfect square plus or minus a constant.

To complete the square for a quadratic expression in terms of \(x\):

• Take an expression like \(x^2 + 4x\).
• Add and subtract the square of half the \(x\) coefficient, \((\frac{4}{2})^2 = 4\), to form a perfect square trinomial.
• This transforms the expression into \((x+2)^2 - 4\).

For equations involving both \(x\) and \(y\), such as our exercise, this process is applied to both terms separately. Completing the square helps us to derive the equation's standard form, making it easier to identify and interpret the type of conic section.

Graphing Conics

Graphing conic sections involves several steps but begins with transforming the equation into its standard form. By doing so, you identify the conic's type: ellipse, parabola, circle, or hyperbola.

In the case of a hyperbola, as in our equation \((x+2)^2 = 4(y-1)^2\), identify key elements:

• The center of the hyperbola is at \((-2, 1)\) which is derived from the transformations applied to \(x\) and \(y\).
• The terms dictate its orientation: horizontally or vertically.
• Specific points like vertices and foci can be plotted using additional numeric constraints from the derived equation.

Using these insights, you can graph the hyperbola, which allows for visual interpretation of its asymptotes and other critical geometric features influencing the overall behavior of the curve.

Equation Transformation

Equation transformation is the series of steps used to convert an equation into a more useful form. This is integral to graphing conics and understanding their properties. In our exercise, we rearranged and simplified the initial equation \(x^{2}-4 y^{2}+4 x+8 y=0\), converting it into a standard form that allows for easy identification and graphing.

The process included:

• Grouping terms with similar variables.
• Using the method of completing the square to simplify each variable group into a perfect square format.
• Rearranging the equation post-completion into a recognizable conic form, specifically for a hyperbola in our case.

These transformations helped identify both the center of the hyperbola and its structural components, such as vertices and asymptotes, to effectively draw the curve. Mastering equation transformations is pivotal in conic section analysis, helping to decipher complex algebraic expressions into geometric insights.