Determine the family of curves represented by the equation y = ax^3 + bx^2 + cx

Determine the family of curves represented by the equation y = ax^3 + bx^2 + cx + d.

Determine the family of curves represented by the equation y = ax^3 + bx^2 + cx + d.

Step 1: Identify the equation given, which is y = ax^3 + bx^2 + cx + d.
Step 2: Recognize that this equation is a polynomial of degree 3, which is called a cubic function.
Step 3: Note that 'a', 'b', 'c', and 'd' are constants that can change, which means they can take different values.
Step 4: Understand that changing the values of 'a', 'b', 'c', and 'd' will create different shapes of the cubic curve.
Step 5: Conclude that all the different shapes created by varying 'a', 'b', 'c', and 'd' represent a family of curves.

Cubic Functions – The equation represents a family of cubic functions characterized by the coefficients a, b, c, and d.
Parameter Variation – The coefficients a, b, c, and d are parameters that can vary, leading to different shapes and positions of the cubic curves.