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In addition to the revision notes for Cubic Graphs on this page, you can also access the following Types of Graphs learning resources for Cubic Graphs
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
|---|---|---|---|---|---|
| 15.3 | Cubic Graphs |
In these revision notes for Cubic Graphs, we cover the following key points:
Cubic graphs bear this name because they are obtained by sketching the set of all points produced by cubic equations in two variables. Such equations are called 'cubic' because the highest power of their independent variable is 3 (given that a cube is a 3-dimensional figure). In arithmetics, they are called "third-degree - or cubic - equations with two variables" while in algebra, they are called "cubic functions".
The general form of cubic functions (equations with two variables) is
where a, b and c are coefficients and d is a constant. The right side of a cubic function represents a third-degree polynomial (i.e. the power of the monomial with the highest degree is 3).
Not all terms may be present in a cubic equation in two variables. You may also encounter cubic equations in two variables in the following forms
The reduced form of a cubic equation with two variables is obtained by substituting y = 0 in the general form. In this way, we obtain
This form helps identify some important points of the graph, such as the x-intercepts. However, to plot the graph we need to consider the corresponding cubic equation with two variables
as it is not sufficient to know only the x-intercepts but many other points of the cubic graph in order to plot it accurately.
Cubic graphs have some features that are specific only to them. Some of these features include:
A common feature of all cubic graphs is a kind of 'wiggle' - a change in the form of the curve that lies around the point where the graph changes shape. A cubic graph may have one or two wiggles, depending on the number of roots the corresponding cubic equation with one variable has. The importance of the wiggles consists of two main elements:
If the cubic graph has a single wiggle, then its central point is a kind of turning point for the graph which produces some symmetry in the sense that, if you rotate by half a cycle one of the 'arms' of the graph, it will fit with the other arm of the same graph.
If the cubic graph has more than one wiggle, then each of these wiggles represents a local minimum or maximum for the corresponding cubic function.
This depends on the sign of coefficient a. Thus, if coefficient a is positive (a > 0), the graph comes from the bottom left and goes up to the top right, as occurred in all cubic graphs seen so far in this tutorial. On the other hand, if coefficient a is negative (a < 0), then the cubic graph comes from the top left and goes down to the bottom right side.
For every x-value in a cubic graph there is a single corresponding y-value. This means cubic graphs have a single y-intercept obtained for x = 0, similar to quadratic or linear graphs. Hence, since the general form of a cubic equation with two variables (cubic function) is
substituting x = 0, we obtain y = d. Therefore, the y-intercept of a cubic function is at M(0, d).
On the other hand, the x-intercepts are obtained for y = 0. In this way, the corresponding equation
is solved either by factorisation or by iterative methods to identify the roots, which are the x-intercepts of the cubic graph.
Not all cubic equations can be factorised. Moreover, even if they can be factorised, one of the expressions in the brackets may not have any real root. This makes most of the corresponding cubic equations have less than three roots.
We are not alwats interested in plotting the graph of a cubic function, especially when the question simply requires to express a rough opinion about the features of the function's graph. In that case, one of the graph features we can determine without needing to plot it consists of the number of wiggles. This number depends on the number of roots the cubic equation has (we called them the number of zeroes), which corresponds to the number of x-intercepts of the graph.
As always, we consider the simplified version of the original cubic function where y is taken as zero. Then, we follow the procedure described below:
Step 1: Remove the constant d from the original equation. This avoids situations where there are wiggles on the same part of the y-values (for example, on the positive part of the Y-axis only) - a situation that makes us impossible to identify them (the wiggles). In this case, you obtain a similar graph to the original but slightly displaced and stretched/compressed.
Step 2: Find the number of roots of the new equation. If it has a single root, the graph has one wiggle and if it has more than one root, the graph has two wiggles.
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