Cubic Function
A number raised to the 3rd power.
A cubic function is a mathematical function of the form f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants. It is called a cubic function because the highest power of x in the function is cube or third power.
A cubic function is characterized by its shape, which is often referred to as a S shape. This shape is determined by the coefficient of the leading term, a. The function has three key features: a y-intercept (where x=0), one or two x-intercepts (depending on the discriminant b² – 4ac), and a turning point (where the function changes from increasing to decreasing, or vice versa).
To graph a cubic function, you can first plot the y-intercept, which is simply the value of d. Then, you can use the x-intercepts and the turning point to help determine the shape of the curve.
The turning point of a cubic function occurs when the derivative of the function is equal to zero. This means that the slope of the curve is neither positive nor negative at that point. The x-coordinate of the turning point can be found by setting the derivative of the function equal to zero and solving for x. Once you have the x-coordinate of the turning point, you can plug it back into the original function to find the y-coordinate.
Cubic functions can be used to model real-world phenomena such as population growth or decay, economic trends, and velocity versus time. They are also important in fields such as engineering, physics, and computer graphics.
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