Final answer:
By completing the square for the x-terms, the equation y = x² - 6x - z² + 9 is transformed into standard form. The resulting equation, (x - 3)² - z² - y = 0, represents a hyperboloid of one sheet.
Step-by-step explanation:
To classify the surface given by the equation y = x² - 6x - z² + 9, we first want to rewrite it in one of the standard forms. To do this, we can complete the square for the x-terms. Completing the square involves finding a number that, when added and subtracted to x² - 6x, turns it into a perfect square trinomial.
We can see that (x - 3)² = x² - 6x + 9 is a perfect square since 3 is half of -6, and 9 is the square of 3. To keep the equation balanced, we'll need to subtract 9 on the right-hand side as well:
y = (x - 3)² - z²
After rearranging slightly, this equation can be recognized as a standard form:
(x - 3)² - z² - y = 0
This is the equation of a hyperboloid of one sheet, which is a standard classification for a quadratic surface.