Problem 5 - Precalculus: Functions and Graphs

Question

Find the vertices, the foci, and the equations of the asymptotes of the hyperbola. Sketch its graph, showing the asymptotes and the foci. $$x^{2}-\frac{y^{2}}{24}=1$$

Step-by-Step Solution

Answer

The vertices are \((-1, 0)\) and \((1, 0)\), the foci are \((-5, 0)\) and \((5, 0)\), and the asymptotes are \(y = \pm 2\sqrt{6}x\).

Step 1: Identify the Form of the Hyperbola

The given equation of the hyperbola is \( x^2 - \frac{y^2}{24} = 1 \). This resembles the standard form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) with \( a^2 = 1 \) and \( b^2 = 24 \). Thus, \( a = 1 \) and \( b = \sqrt{24} = 2\sqrt{6} \). This equation represents a hyperbola that opens horizontally.

Step 2: Determine the Vertices

For a horizontally opening hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are located at \((\pm a, 0)\). With \( a = 1 \), the vertices are at \((-1, 0)\) and \((1, 0)\).

Step 3: Calculate the Foci

The formula to find the foci for the hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) is \((\pm c, 0)\), where \( c^2 = a^2 + b^2 \). Here, \( c = \sqrt{1 + 24} = \sqrt{25} = 5 \). Therefore, the foci are at \((5, 0)\) and \((-5, 0)\).

Step 4: Find the Equations of the Asymptotes

The equations of the asymptotes for a hyperbola in the form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) are given by \( y = \pm \frac{b}{a}x \). Substituting \( a = 1 \) and \( b = 2\sqrt{6} \), the equations are \( y = \pm 2\sqrt{6}x \).

Step 5: Sketch the Graph

To sketch the graph, draw the coordinate axes. Plot the vertices at \((-1, 0)\) and \((1, 0)\), and the foci at \((-5, 0)\) and \((5, 0)\). Sketch the asymptotes by drawing the lines \( y = 2\sqrt{6}x \) and \( y = -2\sqrt{6}x \). Draw the hyperbola branches opening horizontally towards the foci, approaching but never touching the asymptotes.

Key Concepts

Vertices of Hyperbolas

When dealing with hyperbolas, identifying its vertices is a crucial step, as they mark the closest points on the hyperbola to the center. For a hyperbola given by the equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the vertices are positioned at the coordinates \((\pm a, 0)\). This is because the hyperbola opens in the horizontal direction in this specific form. If you take the example from our exercise, \( a = 1 \), the vertices are located at \((-1, 0)\) and \((1, 0)\). Here's a simple way to visualize it:

• Find \( a \) by taking the square root of the denominator under \( x^2 \).


• Place the vertices symmetrically at \((+a, 0)\) and \((-a, 0)\).


• This helps in sketching the hyperbola since the curves of the hyperbola pass through these points.

Vertices not only help in graphing but also in determining the orientation of the hyperbola.

Foci of Hyperbolas

The foci of a hyperbola are points that lie along the axis of symmetry and are crucial for defining the curve's geometric properties. When dealing with a hyperbola such as \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the foci are found using the numbers \( \pm c \), where \( c^2 = a^2 + b^2 \). They lie further from the center compared to the vertices. Let's follow through with our problem. Given:

• \( a = 1 \) and \( b = \sqrt{24} = 2\sqrt{6} \).


• Calculate \( c \) as \( c = \sqrt{1 + 24} = 5 \).


• Therefore, the foci are positioned at \((5, 0)\) and \((-5, 0)\).

Understanding foci is fundamental because these points determine the shape of the hyperbola branches, guiding how the curves open away.

Asymptotes of Hyperbolas

Asymptotes are lines that the branches of a hyperbola approach but never intersect. In hyperbolas, they help approximate the behavior of the curve at large distances from the center. For the equation form \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the asymptotes can be calculated using the equation \( y = \pm \frac{b}{a}x \). Let’s compute them with our example values:

• We have \( a = 1 \) and \( b = 2\sqrt{6} \).


• The asymptote equations become \( y = \pm 2\sqrt{6}x \).

These lines, while not physical parts of the hyperbola, are essential for sketching an accurate graph. They indicate the slope of the hyperbola's branches as they extend to infinity. Recognizing asymptotes ensures a clearer understanding of how the hyperbola behaves far from the origin.