Cubic functions have an equation with the highest power of variable to be 3, i.e. highest power of x is x3.
A function f(x) = x3 has
Domain: {x | } or {x | all real x}
Domain: {y | } or {y | all real y}
We first work out a table of data points, and use these data points to plot a curve:
x | -3 | -2 | -1 | 0 | 1 | 2 | 3 |
y | -27 | -8 | -1 | 0 | 1 | 4 | 27 |
The family of curves f(x) = x3 k can be translated along y-axis by ‘k’ units up or down. For example –
f(x) = x3 + k will be translated by ‘k’ units above the origin, and f(x) = x3 – k will be translated by ‘k’ units below the origin.
The family of curves f(x) = (x k)3 translates the curve y = x3 along the x-axis by ‘k’ units left or right. For example –
f(x) = (x + k)3 will be translated by ‘k’ units towards the left of the origin along the x-axis, and f(x) = (x – k)3 will be translated by ‘k’ units towards the right of the origin along the x-axis.
The domain and range in a cubic graph is always real values.
What type of function is a cubic function?
The function f(x) = x3 increases for all real x, and hence it is a monotonic increasing function (a monotonic function either increases or decreases for all real values of x).
Similarly f(x) = -x3 is a monotonic decreasing function.
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Applying the vertical line test, we can see that the vertical line cuts the curve at only one point. Hence a cubic graph/curve is a function.
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To find out whether it is an odd or an even function, we find out f(-x).
Given f(x) = x3, f'(-x) = (-x)3 = -x3 = -f(x)
f'(-x) = -f(x) means the cubic function f(x) = x3 is an odd function.
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A cubic function of form f(x) = y = x3 has point symmetry.
Example: Sketch the cubic function f(x) = y = x3 + 8.
x-intercept when y = 0 – f(x) = x3 + 8 = 0
x =
= -2. So (-2, 0) is the x-intercept point.
Note: If x3 has a negative value, then the cube root is also negative, because the odd power of negative number is negative.
y-intercept when x = 0 – f(x) = 03 + 8 = 8. So (0, 8) is the y-intercept.
When x =
, y =
, and when x =
, y =
Domain: {all real x}
Range: {all real y}