The family of curves y = kx^3 is known for having:

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Question: The family of curves y = kx^3 is known for having:

Options:

Correct Answer: One turning point

Solution:

The cubic function y = kx^3 has one turning point at x = 0.

The family of curves y = kx^3 is known for having:

The family of curves y = kx^3 is known for having:

Step 1: Understand that the equation y = kx^3 represents a family of cubic functions, where 'k' is a constant that can change the shape of the curve.
Step 2: Recognize that a cubic function can have turning points, which are points where the curve changes direction.
Step 3: To find the turning points, we need to find the derivative of the function y = kx^3.
Step 4: The derivative of y = kx^3 is dy/dx = 3kx^2.
Step 5: Set the derivative equal to zero to find the turning points: 3kx^2 = 0.
Step 6: Solve for x: This equation is zero when x = 0 (since k is a constant and not zero).
Step 7: Conclude that the cubic function y = kx^3 has one turning point at x = 0.

Cubic Functions – Understanding the properties and characteristics of cubic functions, including their turning points and behavior.
Turning Points – Identifying and analyzing turning points in polynomial functions, particularly in cubic functions.