Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
Welcome to our Math lesson on What are the Cubic Graphs?, this is the first lesson of our suite of math lessons covering the topic of Cubic Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Like the other types of graphs discussed so far in our math lessons, 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).
As we stated in chapter 14, we can call equations that contain two variables another name, i.e. 'functions'. (In fact, there is some difference between equations with two variables and functions, consisting of the fact that in equations, all variables have the same importance - no matter which of them you calculate first. This means that, unlike in functions, there are no independent or dependent variables in equations. Recall the systems of linear equations, where for convenience, we sometimes chose to find the variable y before x. Another important difference between equations and functions is that in equations, we are simply interested in calculating the value of an unknown number 'masked' as a variable, while in functions, the focus is to analyse the relationship between the variables. In simpler words, in arithmetics, they are called "third-degree - or cubic - equations with two variables" and in algebra, they are called "cubic functions".)
This clarification done, we can now focus on cubic equations with two variables (or cubic functions if you wish). Thus, since cubic graphs are obtained by raising the independent variable to the third power at maximum, we can write the general form of cubic functions (equations with two variables) as
where a, b and c are coefficients and d is a constant. The right side of a cubic function represents a third-degree polynomial, as those discussed in the 11th chapter of this course.
Obviously, 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
For example,
is a cubic equation in two variables (a cubic function), as the highest power of the variable x is 3. Likewise,
is a cubic equation in two variables (a cubic function), as the highest power of the variable x is 3, despite the latter lack some lower-degree terms.
On the other hand, cubic equations with one variable represent a special case of cubic equations with two variables where the variable y is replaced by a number (usually 0). In this way, we obtain the reduced form of a cubic equation
This form helps identify some important points of the graph, such as the x-intercepts. In math tutorial 9.4, we explained how to find these x-intercepts (we called them 'roots') through iterative methods, such as the half-segment method etc. 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 graph in order to plot it accurately.
You have reached the end of Math lesson 15.3.1 What are the Cubic Graphs?. There are 3 lessons in this physics tutorial covering Cubic Graphs, you can access all the lessons from this tutorial below.
Enjoy the "What are the Cubic Graphs?" math lesson? People who liked the "Cubic Graphs lesson found the following resources useful:
Please provide a rating, it takes seconds and helps us to keep this resource free for all to use
We hope you found this Math tutorial "Cubic Graphs" useful. If you did it would be great if you could spare the time to rate this math tutorial (simply click on the number of stars that match your assessment of this math learning aide) and/or share on social media, this helps us identify popular tutorials and calculators and expand our free learning resources to support our users around the world have free access to expand their knowledge of math and other disciplines.