Exploring the Behavior and Key Points of the Cubic Function y = x^3

y= x^3

cubic parent function

The given equation is y = x^3, which represents a cubic function. In this equation, the variable x represents the input value, and y represents the output value or the function’s value at that input.

A cubic function is a type of polynomial function where the highest power of the variable is 3, in this case, x^3. The graph of a cubic function is a smooth, continuous curve that may be symmetrical or asymmetrical, depending on the coefficients involved.

To understand the behavior of the cubic function y = x^3, let’s look at a few key points:

1. x-intercept: To find the x-intercept (where the graph crosses the x-axis), set y = 0 and solve for x:
0 = x^3
This equation has one real solution, x = 0. So the point (0, 0) is an x-intercept for this cubic graph.

2. y-intercept: To find the y-intercept (where the graph crosses the y-axis), set x = 0:
y = 0^3
This equation yields y = 0 as well. Therefore, the y-intercept is (0, 0).

3. Symmetry: A cubic function may or may not be symmetric. The function y = x^3 is odd, meaning it exhibits symmetry about the origin. In other words, if you reflect the graph across the origin, it will look exactly the same.

4. End behavior: As x approaches negative infinity, y approaches negative infinity, and as x approaches positive infinity, y approaches positive infinity. This indicates that the graph of y = x^3 “spirals” up and down, without bounds.

5. Turning points: In a cubic function like y = x^3, there is exactly one turning point. The turning point of this function is (0, 0).

By considering these points and the shape of the graph, you can sketch the cubic function y = x^3 accurately.

More Answers: