In Exercises 51–56, graph each relation. Use the relation's graph to determine its domain and range. $\frac{x^2}{9}-\frac{y^2}{16}=1$

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1. Recognize that the given equation \(\frac{x^2}{9} - \frac{y^2}{16} = 1\) represents a hyperbola centered at the origin. This is because it matches the standard form of a hyperbola with a horizontal transverse axis: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) where \(a^2 = 9\) and \(b^2 = 16\).
2. To graph the hyperbola, first identify the vertices on the x-axis at \((\pm a, 0)\), which are \((\pm 3, 0)\) in this case. These points are where the hyperbola intersects the x-axis.
3. Next, find the asymptotes of the hyperbola, which are lines that the graph approaches but never touches. For this hyperbola, the asymptotes are given by the equations \(y = \pm \frac{b}{a} x\), or \(y = \pm \frac{4}{3} x\).
4. To determine the domain, consider the values of \(x\) for which the expression under the square root (when solving for \(y\)) is non-negative. Since \(\frac{x^2}{9} - 1 \geq 0\), solve \(\frac{x^2}{9} \geq 1\) which leads to \(|x| \geq 3\). Thus, the domain is \((-\infty, -3] \cup [3, \infty)\).
5. To find the range, solve the equation for \(y\): \(y^2 = 16 \left( \frac{x^2}{9} - 1 \right)\). Since \(y^2 \geq 0\), the range depends on \(x\). For each \(x\) in the domain, \(y\) can take values \(\pm \sqrt{16 \left( \frac{x^2}{9} - 1 \right)}\). Therefore, the range is all real numbers except values between \(-0\) and \(0\) when \(|x| < 3\), so the range is \((-\infty, \infty)\) but with \(y=0\) only at \(x=\pm 3\).

Key Concepts

Hyperbola Equation and Its Graph

The given equation represents a hyperbola, a type of conic section characterized by two separate branches. It is defined by the difference of squared terms divided by constants equal to 1. Understanding the standard form helps in sketching the graph, identifying vertices, asymptotes, and the orientation of the hyperbola.

Domain of a Relation

The domain of a relation is the set of all possible x-values for which the relation is defined. For the hyperbola, the domain is restricted by the values of x that keep the expression under the square root or denominator valid, ensuring the equation holds true and the graph exists.

Range of a Relation

The range is the set of all possible y-values that the relation can take. For the hyperbola, the range depends on the values of y that satisfy the equation for given x-values, often found by analyzing the graph or solving the equation for y in terms of x.