Problem 53 - A Graphical Approach to College Algebra

Question

Graph each hyberbola by hand. Give the domain and range. Do not use a calculator. $$9 x^{2}-4 y^{2}=1$$

Step-by-Step Solution

Answer

The domain is \(( -\infty, \infty)\). The range is \((-\infty, -\frac{1}{2}) \cup (\frac{1}{2}, \infty)\).

Step 1: Identify the Hyperbola Form

The given equation is \(9x^2 - 4y^2 = 1\). This is in the standard form of a hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Thus, we can identify \(a^2 = \frac{1}{9}\) and \(b^2 = \frac{1}{4}\).

Step 2: Calculate a and b

From \(a^2 = \frac{1}{9}\), we find \(a = \frac{1}{3}\). From \(b^2 = \frac{1}{4}\), we get \(b = \frac{1}{2}\). These values will help us plot the vertices and asymptotes.

Step 3: Plot the Center, Vertices and Asymptotes

The hyperbola is centered at the origin \((0, 0)\). The vertices are located at \((\pm a, 0)\), which are \((\pm \frac{1}{3}, 0)\). The asymptotes can be drawn using the slope \(\pm \frac{b}{a} = \pm \frac{1/2}{1/3} = \pm \frac{3}{2}\), passing through the origin.

Step 4: Sketch the Hyperbola

With the vertices and asymptotes plotted, draw the branches of the hyperbola opening along the x-axis. The branches should approach but never intersect the asymptotes.

Step 5: Define the Domain and Range

For the hyperbola centered at the origin horizontal to the x-axis, the domain is all real numbers, \(( -\infty, \infty)\). The range is limited to the y-values where the hyperbola exists: \((-\infty, -\frac{1}{2}) \cup (\frac{1}{2}, \infty)\).

Key Concepts

Standard Form of Hyperbola

In mathematics, understanding the standard form of a hyperbola is crucial for graphing and analysis. A hyperbola is a set of all points \(x, y\) such that the difference of their distances to two fixed points called foci is constant. The standard form of a hyperbola can be written as:

• Horizontal hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\)
• Vertical hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\)

In the equation \(9x^2 - 4y^2 = 1\), the coefficients of \(x^2\) and \(y^2\) help determine whether it is a horizontal or vertical hyperbola. Here, \(x^2\) has a positive coefficient, indicating a horizontal hyperbola. Transforming the equation into the standard form involves rewriting it as \(\frac{x^2}{\left(\frac{1}{3}\right)^2} - \frac{y^2}{\left(\frac{1}{2}\right)^2} = 1\). The values \(a\) and \(b\) are found by taking the square roots of the denominators: \(a=\frac{1}{3}\) and \(b=\frac{1}{2}\). This structure helps in sketching its graph by identifying vertex locations and asymptote slopes.

Domain and Range of Hyperbola

The domain and range of a hyperbola describe the set of all possible x-values (domain) and y-values (range) that comprise the graph of the hyperbola. For the equation \(9x^2 - 4y^2 = 1\), we have a horizontal hyperbola centered at the origin \( (0,0)\). Understanding the shape and orientation of the hyperbola is essential in identifying its domain and range.For any hyperbola of the form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), the domain is all real numbers \( (-\infty, \infty)\) because the graph extends infinitely along the x-axis. For the range, we have two separate intervals where the hyperbola exists, reflecting its two branches. Here the branches occur above and below the x-axis, excluding the interval from \(-\frac{1}{2}\) to \(\frac{1}{2}\). Therefore, the range is \((-\infty, -\frac{1}{2}) \cup (\frac{1}{2}, \infty)\). This means y can be any value below \(-\frac{1}{2}\) or above \(\frac{1}{2}\). Recognizing these intervals aids in completing accurate graphs by hand.

Graphing Hyperbolas by Hand

Graphing a hyperbola by hand can seem daunting, but breaking it down into clear steps makes it manageable. Start by using the standard form \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) to identify two vital components: the center and the vertices.

• First, identify the center, which in this exercise is \( (0,0)\), making it your starting point.
• Next, use the value of \(a\) to mark the vertices along the x-axis at \( (\pm \frac{1}{3}, 0)\).
• The asymptotes, lines that the hyperbola approaches but never touches, play a key role in drawing the graph's framework. They have a slope of \(\pm \frac{b}{a}\). Here, the asymptotes slope \(\pm \frac{3}{2}\).
• Draw the asymptotes through the center with the calculated slope.
• Finally, sketch the hyperbola using the vertices and asymptotes as guides, forming two distinct curves that mirror each other along these lines. These curves open horizontally along the x-axis, approaching the asymptotes but not touching them.

This careful plotting ensures your hand-drawn graph accurately reflects the hyperbola's mathematical form.