Identify the family of curves represented by the equation y = c/x, where c is a

Identify the family of curves represented by the equation y = c/x, where c is a constant.

Identify the family of curves represented by the equation y = c/x, where c is a constant.

Correct Answer: Hyperbolas

Step 1: Understand the equation y = c/x. Here, 'c' is a constant that can take different values.
Step 2: Recognize that this equation can be rewritten as xy = c, which is a standard form of a hyperbola.
Step 3: Identify that as 'c' changes, the shape and position of the hyperbola changes, but they all belong to the same family of curves.
Step 4: Note that hyperbolas have asymptotes, which are lines that the curve approaches but never touches. For this equation, the asymptotes are the x-axis (y=0) and the y-axis (x=0).
Step 5: Conclude that the equation y = c/x represents a family of hyperbolas, each with the same asymptotes but different sizes and orientations depending on the value of 'c'.

Family of Curves – The equation y = c/x represents a family of hyperbolas, where each curve corresponds to a different value of the constant c.
Asymptotes – The hyperbolas have asymptotes along the x-axis and y-axis, which are important characteristics of their shape.