Problem 10 - Precalculus: Mathematical for Calculus

Question

Find the center, foci, vertices, and asymptotes of the hyperbola. Then sketch the graph. $$(x-8)^{2}-(y+6)^{2}=1$$

Step-by-Step Solution

Answer

Center: (8, -6), Vertices: (7, -6) and (9, -6), Foci: (8±√2, -6), Asymptotes: y=x-14, y=-x+2.

Step 1: Identify the Hyperbola Form

The given equation \((x-8)^2 - (y+6)^2 = 1\) is in the standard form of a hyperbola: \((x-h)^2 - (y-k)^2 = 1\). This indicates a hyperbola centered at \((h, k)\) with a horizontal transverse axis.

Step 2: Find the Center

From \((x-h)^2 - (y-k)^2 = 1\), we identify \(h = 8\) and \(k = -6\). So the center of the hyperbola is \((8, -6)\).

Step 3: Calculate the Vertices

For a hyperbola \((x-h)^2 - (y-k)^2 = 1\), the vertices on the transverse axis are at \((h \,\pm\, a, k)\). In this case, \(a = 1\) (since the denominator under \((x-h)^2\) is 1). Thus, the vertices are \((7, -6)\) and \((9, -6)\).

Step 4: Find the Foci

The distance of the foci from the center is \(c\) where \(c^2 = a^2 + b^2\). Here, \(c = \sqrt{1+1} = \sqrt{2}\). The foci are at \((8 \pm \sqrt{2}, -6)\).

Step 5: Determine the Asymptotes

For the given form, the asymptotes are lines \(y = k \pm \frac{b}{a}(x-h)\). Here \(a = 1\) and \(b = 1\), so the equations are \(y = -6 \pm (x-8)\). This simplifies to \(y = x - 14\) and \(y = -x + 2\).

Step 6: Sketch the Hyperbola

Plot the center \((8, -6)\), vertices \((7, -6)\) and \((9, -6)\), and foci \((8 \pm \sqrt{2}, -6)\). Draw the asymptotes: lines with equations \(y = x - 14\) and \(y = -x + 2\). For a horizontal hyperbola, sketch the hyperbola opening left and right through the vertices.

Key Concepts

Conic Sections

Conic sections are shapes created by the intersection of a cone and a plane. They encompass four basic shapes: circles, ellipses, parabolas, and hyperbolas. Each conic section has unique characteristics defined by specific equations. In mathematics, understanding conic sections is crucial for exploring geometric properties and relationships.

• Circles: This is a special ellipse where both foci are at the center.
• Ellipses: They have a longer and shorter axis with two foci inside the ellipse.
• Parabolas: These are U-shaped and open either upwards, downwards, or sideways.
• Hyperbolas: These come as two separate pieces or branches and are defined by a key equation structure.

Hyperbolas differ from other conic sections because they consist of two disjoint curves and open either horizontally or vertically. In-depth knowledge of their construction and properties helps in graphing and solving hyperbolic equations.

Vertices and Foci of Hyperbolas

The vertices and foci are vital points that define a hyperbola's shape and orientation. For hyperbolas, the vertices are the points where each branch of the hyperbola is closest to each other.
A hyperbola’s equation in standard form \[ (x-h)^2/a^2 - (y-k)^2/b^2 = 1 \] shows the vertices and foci positioning. In this equation:

• Vertices: For a horizontal hyperbola, the vertices are determined using \((h\pm a, k)\)\
• Foci: They are at \((h \pm c, k)\), where \(c = \sqrt{a^2 + b^2}\). Foci are located further from the center than the vertices, one on each side.

Knowing the distances and using the right formulae helps in finding these key points quickly for any given hyperbola.

Graphing Hyperbolas

Graphing a hyperbola may initially seem complex, but understanding its structure simplifies the task. Start by identifying the hyperbola’s center point; this acts as the reference for plotting other critical components.
Once the center is known, drawing the vertices on the line of the transverse axis helps set the orientation and width of the graph. Here's how the process flows:

• Plot the center and vertices on a graph as initial markers.
• Calculate and plot the foci on either side beyond the vertices.
• Draw asymptotes—guide lines which the hyperbola approaches but never touches.
• Sketch the hyperbola's branches, ensuring to bend them toward the asymptotes.

This approach visualizes what would otherwise be an abstract equation, making it easier to interpret and reenact for all hyperbolas.

Asymptotes

Asymptotes are lines that hyperbolas draw near but never cross. They act as guides to form the shape of the hyperbola.
For hyperbolas represented by the standard equation\[(x-h)^2/a^2 - (y-k)^2/b^2 = 1\],the asymptotes have equations derived from the center: \[(y-k) = \pm \frac{b}{a}(x-h).\]

• Horizontal Hyperbolas: For these, the asymptotes are crossed with a positive and a negative slope.
• Vertical Hyperbolas: Here, the orientation changes, but the formula usage remains the same.

Asymptotes assist in defining the "opening" path that the hyperbola will take. Consistently using them while graphing ensures that the hyperbola's curvature progresses correctly towards infinity.