Determine the family of curves represented by the equation y = k/x, where k is a

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Question: Determine the family of curves represented by the equation y = k/x, where k is a constant.

Options:

Correct Answer: Hyperbolas

Solution:

The equation y = k/x represents a family of hyperbolas with varying values of \'k\'.

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

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

Step 1: Understand the equation y = k/x. Here, 'y' is the output, 'k' is a constant, and 'x' is the input.
Step 2: Recognize that 'k' can take any value (positive, negative, or zero), which will change the shape of the curve.
Step 3: Rewrite the equation in a different form: y * x = k. This shows that the product of 'y' and 'x' is constant (equal to 'k').
Step 4: Identify that as 'k' changes, the curves will shift and stretch, but they will all have a similar shape, which is a hyperbola.
Step 5: Conclude that the equation y = k/x represents a family of hyperbolas, each corresponding to a different value of 'k'.

Family of Curves – The equation y = k/x represents a family of hyperbolas, where each curve corresponds to a different value of the constant k.
Hyperbolas – The shape of the curves is hyperbolic, which is characterized by the product of the coordinates being constant.
Parameter Variation – The constant k acts as a parameter that shifts the hyperbola along the axes, affecting its position but not its shape.