Problem 54 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. The problems include equations that describe circles, parabolas, ellipses, and hyperbolas. $$ \frac{y^{2}}{36}-\frac{(x+2)^{2}}{4}=1 $$

Step-by-Step Solution

Answer

The hyperbola is vertically oriented with center at (-2, 0), vertices at (-2, 6) and (-2, -6), and asymptotes y = ±3(x + 2).

Step 1: Identify the Type of Conic Section

The given equation \( \frac{y^2}{36} - \frac{(x+2)^2}{4} = 1 \) is in the form \( \frac{y^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\). Comparing this with the standard form of a hyperbola \( \frac{y^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \), we identify that this is the equation of a vertically oriented hyperbola.

Step 2: Write the Equation in Standard Form

The equation \( \frac{y^2}{36} - \frac{(x+2)^2}{4} = 1 \) is already in standard form for a hyperbola, being \( \frac{y^2}{a^2} - \frac{(x-h)^2}{b^2} = 1 \). This aligns perfectly as \( a^2 = 36 \) and \( b^2 = 4 \), with \( h = -2 \).

Step 3: Determine the Hyperbola Parameters

For the given hyperbola, \( a^2 = 36 \) implies \( a = 6 \) and \( b^2 = 4 \) implies \( b = 2 \). The center is at \((h, k) = (-2, 0)\). The vertices are located at \((h, k \pm a)\), giving \((-2, 0 \pm 6)\), or the points \((-2, 6)\) and \((-2, -6)\).

Step 4: Plot the Asymptotes

Asymptotes for a vertical hyperbola can be found using the equations \( y = k \pm \frac{a}{b}(x-h) \). With \( k = 0 \), \( a = 6 \), \( b = 2 \), and \( h = -2 \), the equations of the asymptotes are \( y = \pm 3(x + 2) \).

Step 5: Graph the Hyperbola

Plot the center at \((-2, 0)\), and draw the vertices \((-2, 6)\) and \((-2, -6)\). Then, sketch the asymptotes using the equations \( y = 3(x + 2) \) and \( y = -3(x + 2) \). Draw the branches of the hyperbola approaching these asymptotes and passing through the vertices.

Key Concepts

Hyperbola

A hyperbola is one of the conic sections, which is formed from the intersection of a double cone with a plane. Unlike circles and ellipses, which are defined by the sum of distances from two points being constant, a hyperbola is defined by the difference.
Hyperbolas consist of two disconnected curves called branches, which mirror each other and open either side by side or up and down. These branches tend to extend infinitely without touching but appear as if they are coming closer as viewed along their paths. They are symmetric about their center and axes, creating an interesting visual and analytic study field.
Understanding hyperbolas, especially through graphing, helps to recognize patterns and solve related algebraic equations effectively.

Standard Form of Equation

The standard form of a hyperbola's equation helps simplify the process of identifying its components and graphing it. For a vertically oriented hyperbola, the standard form is given by:
\[\frac{y^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\]Here, \(a\) and \(b\) are real numbers derived from the equation’s denominators. The values of \(a^2\) and \(b^2\) are crucial in determining the shape and size of the hyperbola. The center of the hyperbola is positioned at the coordinates \( (h, k) \), extracted directly from the equation.

• \(a^2\) relates to the distance from the center to the vertices along the y-axis.
• \(b^2\) affects the slope of the asymptotes.

Having the hyperbola in this standard form efficiently guides the correct plotting of its curve and elements, allowing a clear visual representation to emerge from its algebraic definition.

Graphing Equations

Graphing a hyperbola involves plotting several key elements on the coordinate plane, leading to an accurate depiction of its form. These elements include:

• Center: A reference point from which distances to other parts of the graph are measured. For the sample equation, the center is at \((-2, 0)\).
• Vertices: These are points on the hyperbola located at the distance \(a\) from the center, along the hyperbola’s axis. Calculate them as \( (h, k \pm a) \).
• Co-Vertices: Not as crucial for graphing, but these are found on the secondary axis, affecting the square region of asymptotes.
• Asymptotes: Lines toward which the hyperbola’s branches approach.

Once the center and vertices are plotted, sketching asymptotes provides a guideline, aligning the curves of the hyperbola correctly for a proper sketch. Drawing the branches requires ensuring that they approach the asymptotes symmetrically, highlighting the hyperbola’s balanced form.

Asymptotes

Asymptotes are crucial linear guides that aid in sketching a hyperbola. They are straight lines that the branches of a hyperbola approach but never actually meet. In mathematical terms, they are the lines that the curve infinitely approaches.
For the given hyperbola equation:
\[y = k \pm \frac{a}{b}(x - h)\]
This provides their equations. The asymptotes derive their slope from \( \frac{a}{b} \), influencing how steep or shallow they appear. Here, substituting \(a = 6 \), \(b = 2 \), \(h = -2\), and \(k = 0\) into the equation yields:

• \(y = 3(x + 2)\)
• \(y = -3(x + 2)\)

Plot these on the graph as guide lines intersecting at the center, assisting in accurate curve drawing of the hyperbola. They illustrate the directional nature, around which the hyperbola twists without ever crossing or touching.