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

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1. Recognize that the given equation \(\frac{y^2}{16} - \frac{x^2}{9} = 1\) represents a hyperbola centered at the origin. This is because it matches the standard form of a vertical hyperbola: \(\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1\) where \(a^2 = 16\) and \(b^2 = 9\).
2. Identify the values of \(a\) and \(b\) by taking square roots: \(a = 4\) and \(b = 3\). These values help determine the shape and orientation of the hyperbola.
3. To graph the hyperbola, plot the vertices on the \(y\)-axis at \((0, \pm a)\), which are \((0, \pm 4)\). These points are where the hyperbola intersects the \(y\)-axis.
4. Determine the domain by solving for \(x\) in terms of \(y\). Rearrange the equation to isolate \(x^2\): \(\frac{y^2}{16} - 1 = \frac{x^2}{9}\). Then multiply both sides by 9 to get \(x^2 = 9\left(\frac{y^2}{16} - 1\right)\). For \(x^2\) to be non-negative (real \(x\) values), the expression inside the parentheses must be greater than or equal to zero. Use this to find the range of \(y\) values that produce real \(x\) values, which will help determine the domain of the relation.
5. Similarly, determine the range by considering the possible \(y\) values. Since the hyperbola opens vertically, \(y\) can take any real value such that the expression under the square root for \(x\) is valid. Use the inequality from the previous step to find the exact range of \(y\) values.

Key Concepts

Hyperbola Equation and Standard Form

A hyperbola is a conic section defined by an equation where the difference of squared terms equals a constant, such as \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \). This form indicates a vertical transverse axis, with vertices at \( (0, \pm a) \). Understanding this standard form helps in identifying the shape and orientation of the graph.

Domain and Range of Relations

The domain of a relation is the set of all possible x-values, while the range is the set of all possible y-values. For hyperbolas, these sets depend on the equation's restrictions and asymptotes. Determining domain and range from the graph involves analyzing where the curve exists along the x- and y-axes.

Graphing Conic Sections

Graphing conic sections like hyperbolas requires plotting key points such as vertices and asymptotes, which guide the curve's shape. For the given equation, asymptotes are lines that the hyperbola approaches but never touches, found by setting the equation equal to zero. Accurate graphing aids in visualizing the relation and extracting domain and range.