Cubic Function
Interesting Facts About Cubic Functions:
* The solution to the cubic was published by Gerolamo Cardano (1501-1576) in his treatise Ars Magna. However, Cardano was not the original discoverer of either of these results.
* Cubic Functions can be used to model three-dimensional objects to allow you to identify a missing dimension or explore the result of changes to one or more dimensions.
Example
A cubic function is one in the form f(x)=ax^3+bx^2+cx+d
The "basic" cubic function, f(x)=x^3 is graphed below.
The function of the coefficient a in the general equation is to make the graph "wider" or "skinnier", or to reflect it (if negative):
The constant d in the equation is the y -intercept of the graph.
The effects of b and c on the graph are more complicated. However, if you can factor the right side of the equation, you can find one or more x -intercepts , and use these to sketch the graph. (Some cubics, however, cannot be factored.)
A cubic function may have one, two or three x-intercepts, corresponding to the real roots of the related cubic equation.
How to graph cubic functions by plotting points?
The general form of a cubic function is y = ax^3 + bx^2 + cx + d where a , b, c and d are real numbers and a is not zero.
We can graph cubic functions by plotting points.
Example:
Draw the graph of y = x^3 + 3 for -3 ≤ x ≤ 3. Use your graph to find:
a) the value of y when x = 2.5
b) the value of x when y = -15
Solution:
a) When x = 2.5, y =18.6
b) When y = -15, x =-2.6
Graph this on a graphing calculator to see the function:
Examples:
Let x represent the side length of each square cut out of a corner, then the dimensions of the box will be 8−2x, 8−2x and x, for length, width, and height, respectively.
(8-2x)(8-2x)x = 4x^3-32x^2+64x
The volume of the box as a function of side length:
4x^3-32x^2+64x
- What is the maximum volume of an open-top box, made from a square piece of cardboard with an area of 64 ft^2?(solve for y)
The maximum volume is approximately 38 ft^3
2. Assuming the box is made by cutting squares from each corner to allow the sides to fold up, what are the dimensions of those square cut-outs?(solve for x)
The square corners are approximately 1.3 ft on each side