Use graphing software to graph the functions specified in Exercises 31–36.
Select a viewing window that reveals the key features of the function.


Graph the upper branch of the hyperbola y² − 16x² = 1.

Verified step by step guidance

1. Identify the equation of the hyperbola: \( y^2 - 16x^2 = 1 \). This is a standard form of a hyperbola centered at the origin with a vertical transverse axis.
2. Rewrite the equation in the standard form for a hyperbola: \( \frac{y^2}{1} - \frac{x^2}{\frac{1}{16}} = 1 \). This shows that \( a^2 = 1 \) and \( b^2 = \frac{1}{16} \).
3. Determine the vertices and asymptotes: The vertices are at \( (0, \pm a) = (0, \pm 1) \). The equations of the asymptotes are \( y = \pm \frac{1}{4}x \).
4. Select a suitable viewing window for graphing: Choose a window that includes the vertices and shows the asymptotic behavior. A possible window could be \([-2, 2]\) for \(x\) and \([-2, 2]\) for \(y\).
5. Use graphing software to plot the upper branch of the hyperbola: Input the equation \( y = \sqrt{1 + 16x^2} \) to graph the upper branch, ensuring the graph is within the selected viewing window.

Key Concepts

Hyperbola

A hyperbola is a type of conic section formed by the intersection of a plane with both nappes of a double cone. It consists of two disconnected curves called branches. The standard form of a hyperbola centered at the origin is x²/a² - y²/b² = 1 or y²/a² - x²/b² = 1, where a and b are real numbers that determine the shape and orientation of the hyperbola.

Graphing Functions

Graphing functions involves plotting points on a coordinate plane to visualize the behavior of a function. Key features to consider include intercepts, asymptotes, and the general shape of the graph. For a hyperbola, it's important to identify the vertices, foci, and asymptotes to accurately represent its branches.

Viewing Window

The viewing window in graphing software determines the portion of the coordinate plane visible on the screen. Selecting an appropriate viewing window is crucial for capturing the key features of a function, such as intercepts and asymptotes, and ensuring that the graph is neither too zoomed in nor too zoomed out, which can obscure important details.