Problem 10 - Precalculus Mathematics for Calculus

Question

Find the vertices, foci, and asymptotes of the hyperbola, and sketch its graph. $$\frac{y^{2}}{9}-\frac{x^{2}}{16}=1$$

Step-by-Step Solution

Answer

Vertices: (0, 3), (0, -3). Foci: (0, 5), (0, -5). Asymptotes: \(y=\pm\frac{3}{4}x\).

Step 1: Identify the Conic Type and Standard Form

The given equation is \( \frac{y^{2}}{9} - \frac{x^{2}}{16} = 1 \), which is in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This represents a vertically oriented hyperbola, where the positive term is the \( y^2 \) term.

Step 2: Determine the Values of \( a \) and \( b \)

From the equation, we identify that \( a^2 = 9 \) and \( b^2 = 16 \). Therefore, \( a = 3 \) and \( b = 4 \).

Step 3: Find the Vertices

For a vertically oriented hyperbola, the vertices are located at \((0, \pm a)\). Hence, the vertices are \((0, 3)\) and \((0, -3)\).

Step 4: Calculate the Foci

The foci for a vertically oriented hyperbola are found using the formula \( c^2 = a^2 + b^2 \). Calculating \( c \), we find: \( c^2 = 9 + 16 = 25 \) \( c = 5 \). Thus, the foci are \((0, \pm 5)\), or \((0, 5)\) and \((0, -5)\).

Step 5: Determine the Asymptotes

The equations for the asymptotes of a vertical hyperbola are \( y = \pm\frac{a}{b}x \). Substituting the values of \( a \) and \( b \), we get \( y = \pm\frac{3}{4}x \).

Step 6: Sketch the Hyperbola

The graph of the hyperbola is centered at the origin. It opens upwards and downwards because the \( y^2 \) term is positive. Sketch the vertices \((0, 3)\) and \((0, -3)\), the foci \((0, 5)\) and \((0, -5)\), and include the asymptotes \( y = \frac{3}{4}x \) and \( y = -\frac{3}{4}x \) as dashed lines.

Key Concepts

Vertices of Hyperbola

The vertices of a hyperbola play a vital role in understanding its geometry. In a hyperbola, there are two vertices, which lie on the axis of symmetry, and they help define the "endpoints" of each branch of the hyperbola. For the given vertically oriented hyperbola equation \[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \] the vertices are determined based on the value of \( a \), where \( a^2 = 9 \). Working through this, we find \( a = 3 \). Since the hyperbola opens vertically, the vertices are located at

• \( (0, a) = (0, 3) \)
• \( (0, -a) = (0, -3) \)

These points show where the hyperbola stretches along the y-axis. To clearly visualize the hyperbola, plotting these points on a graph is essential. This gives us better insight into its orientation and size.

Foci of Hyperbola

The foci are another critical element of a hyperbola, providing a deeper look into its structure. For hyperbolas, the distance to each focus influences the curve of the shape. When considering our vertically-oriented hyperbola:\[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \] we calculate the foci using the formula \( c^2 = a^2 + b^2 \). Here, \( a^2 = 9 \) and \( b^2 = 16 \), giving us: \[ c^2 = 9 + 16 = 25 \] \[ c = 5 \] Thus, the foci are positioned along the vertical y-axis at

• \( (0, c) = (0, 5) \)
• \( (0, -c) = (0, -5) \)

The foci determine the major axis of the hyperbola, which is the longest line running through its center, stretching from one focus to the other. Observing where these foci lie, you can better understand the overall direction and stretch of the hyperbola.

Asymptotes of Hyperbola

Asymptotes are crucial for sketching hyperbolas. They are imaginary lines that the hyperbola approaches as the points move further from the center. These lines help define the hyperbola's directional flow toward infinity and provide a guide for drawing the curve.For our hyperbola \[ \frac{y^2}{9} - \frac{x^2}{16} = 1 \] with a vertical orientation, the asymptotes are calculated using the slope \( \pm \frac{a}{b} \) from the equation. Here,

• \( a = 3 \)
• \( b = 4 \)

The equations for the asymptotes are then given by:

• \( y = \frac{3}{4}x \)
• \( y = -\frac{3}{4}x \)

These diagonal lines lead the eye through the center of the hyperbola and continue outward in a diagonal direction. By marking out these asymptotes as dashed lines on a graph, they serve as boundaries that the hyperbola will never cross, shaping the curve's pathway. Understanding asymptotes ensures a more accurate sketch of the hyperbola's infinite branches.