Understanding Cubic and Quartic Graphs

A cubic function is an algebraic function of the third degree. The general form is f(x) = ax³ + bx² + cx + d, where a, b, c, and d are constants and a ≠ 0.
A quartic function is a function of the fourth degree. The general form is f(x) = ax⁴ + bx³ + cx² + dx + e, where a, b, c, d, and e are constants and a ≠ 0.
The degree of the function determines its highest power and influences the graph’s shape and the number of solutions the equation may have.
Cubic functions can have various graph shapes, but the end behaviour of all cubic functions will have opposite directions.
Quartic functions can have an even wider range of shapes but always exhibit the same end behaviour; that is, the ends of the graph either both point upwards or both point downwards.

Graph Characteristics

Roots or zeros of the function are the x-values where the function crosses or touches the x-axis.
When the function touches but does not cross the x-axis, that root is sometimes called a double root or repeated root.
The y-intercept is the point where the graph crosses the y-axis. This is always the constant term of the function when set to f(x) = y.
The turning points or extrema of the function are the high and low points of the graph. For cubic functions, there can be either one or two extrema, while quartic can have up to three extrema.
For quartic functions, there is also a point of inflection, which is a point on the graph where the curve changes from being concave up to concave down, or vice versa.

To sketch a cubic or quartic graph accurately, identify the roots and the y-intercept.
For the cubic, determine if the function is positive or negative. If the highest degree coefficient is positive, the right side of your graph will point upwards. If it is negative, it will point downwards.
For a quartic function, both ends point in the same direction. Sketch the graph accordingly.
Plot the turning points and points of inflection to provide an accurate representation of the graph.
Join the points smoothly, noting that cubic and quartic functions are continuous and will have no breaks or gaps.

Understanding cubic and quartic graphs enhances comprehension of more complex algebraic concepts.
These graphs demonstrate the effect of higher powers on polynomial functions, helping to create a better understanding of the behaviour of more complex mathematical models.
This knowledge assists in solving more advanced mathematical problems by simplifying them into graphical interpretations.
The ability to sketch and interpret cubic and quartic graphs is an important skill in both pure mathematics and practical engineering, physics, and computer science applications.