Final answer:
To sketch a hyperbola, find its equation in standard form, identify the center, vertices, and foci, draw a central rectangle and asymptotes, plot the vertices, and finally sketch the hyperbola branches approaching the asymptotes.
Step-by-step explanation:
To sketch the graph of a hyperbola, one must follow a series of steps. A hyperbola is a set of points in a plane such that the difference of the distances from any point on the curve to two fixed points (the foci) is constant. Here is the process:
- First, determine the equation of the hyperbola in standard form. For example, x^2/a^2 - y^2/b^2 = 1 where a and b are real numbers.
- Identify the center, vertices, and foci of the hyperbola. The center is at the origin (h, k), vertices are (h ± a, k) and foci are (h ± c, k) where c^2 = a^2 + b^2.
- Draw the rectangle centered at the hyperbola's center, with dimensions 2a and 2b, then draw the asymptotes by connecting opposite corners of this rectangle.
- Plot the vertices (points where the hyperbola crosses the axes).
- Finally, sketch the two branches of the hyperbola, making sure they approach the asymptotes as they move away from the center.
This standard form helps in clearly identifying essential components needed to draw an accurate sketch.