Identify the family of curves represented by the equation y^2 = 4ax.

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Question: Identify the family of curves represented by the equation y^2 = 4ax.

Options:

Correct Answer: Parabolas

Solution:

The equation y^2 = 4ax represents a family of parabolas that open to the right with varying values of \'a\'.

Identify the family of curves represented by the equation y^2 = 4ax.

Identify the family of curves represented by the equation y^2 = 4ax.

Correct Answer: Parabolas opening to the right

Step 1: Start with the equation y^2 = 4ax.
Step 2: Recognize that this is a quadratic equation in terms of y.
Step 3: Identify the standard form of a parabola, which is y^2 = 4px, where p is a constant.
Step 4: In our equation, 4a is equivalent to 4p, meaning a = p.
Step 5: Understand that 'a' can take any positive value, which means the shape of the parabola changes.
Step 6: Conclude that the equation represents a family of parabolas that all open to the right.

Parabolas – The equation y^2 = 4ax describes parabolas that open to the right, where 'a' determines the width and position of the parabola.