Cubic function

Definition

A cubic function is a polynomial function of degree three, represented in the form $$f(x) = ax^3 + bx^2 + cx + d$$, where 'a', 'b', 'c', and 'd' are constants and 'a' is not zero. Cubic functions can model a variety of real-world phenomena and are characterized by their distinct 'S' shaped graphs, which can have one or two turning points depending on the nature of the coefficients.

5 Must Know Facts For Your Next Test

1. The graph of a cubic function can intersect the x-axis up to three times, indicating up to three real roots.
2. Cubic functions can have either one real root and two complex roots or three real roots, depending on their discriminant.
3. The coefficients of a cubic function influence its shape, such as the steepness and direction of its curves.
4. Cubic functions have at most two turning points, which can be found using calculus or by analyzing the first derivative.
5. Cubic functions exhibit symmetry if they are expressed in a certain form, such as when the middle terms cancel out, creating a simpler visual representation.

Review Questions

• How does the degree of a cubic function affect its graph compared to linear and quadratic functions?
  • The degree of a cubic function, which is three, allows for more complex behavior in its graph than linear (degree one) and quadratic (degree two) functions. While linear functions produce straight lines and quadratic functions create parabolas with one turning point, cubic functions can feature an 'S' shape with up to two turning points. This complexity allows cubic functions to better model situations where changes occur in different directions.
• In what ways can you identify the turning points of a cubic function graphically and algebraically?
  • Turning points of a cubic function can be identified graphically by observing where the curve changes direction, which corresponds to local maxima and minima. Algebraically, these points can be found by calculating the first derivative of the cubic function and setting it equal to zero. Solving this equation yields potential x-values for turning points, which can then be tested with the second derivative to confirm whether each point is a maximum or minimum.
• Evaluate how changing the coefficients in a cubic function impacts its end behavior and overall shape.
  • Changing the coefficients in a cubic function affects both its end behavior and overall shape significantly. The leading coefficient determines whether the ends of the graph rise or fall; if it's positive, both ends will rise to infinity, while a negative coefficient will result in one end rising and the other falling. Additionally, adjusting other coefficients influences the location and number of turning points as well as how steeply or gently the graph transitions between these points, leading to various unique shapes.

Related terms

Polynomial function: A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients.

Turning point: A turning point is a point on the graph of a function where the direction of the curve changes, indicating local maxima or minima.

End behavior: End behavior describes how the values of a function behave as the input values approach positive or negative infinity, which is particularly important for polynomial functions.