key term - Cubic

Definition

A cubic function is a polynomial function of degree three, meaning it contains a variable raised to the power of three. These functions have a distinctive S-shaped curve and can be used to model a wide range of real-world phenomena, including population growth, projectile motion, and the shape of certain architectural structures.

5 Must Know Facts For Your Next Test

1. The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are real numbers.
2. Cubic functions can have up to three real roots, which are the values of $x$ where the function equals zero.
3. The graph of a cubic function is an S-shaped curve that may have one or more points of inflection, where the concavity of the curve changes.
4. Cubic functions can be used to model a variety of real-world phenomena, such as the growth of a population, the trajectory of a projectile, or the shape of a suspension bridge.
5. Dividing polynomials, including cubic polynomials, can be done using the long division algorithm or the synthetic division algorithm.

Review Questions

• Explain the general form of a cubic function and describe the key features of its graph.
  • The general form of a cubic function is $f(x) = ax^3 + bx^2 + cx + d$, where $a$, $b$, $c$, and $d$ are real numbers. The graph of a cubic function is an S-shaped curve that may have up to three real roots, where the function equals zero. The graph can also have one or more points of inflection, where the concavity of the curve changes. These key features allow cubic functions to model a wide range of real-world phenomena, such as population growth, projectile motion, and the shape of certain architectural structures.
• Describe the relationship between cubic functions and the process of dividing polynomials.
  • Cubic functions, as a type of polynomial function, can be divided using the long division algorithm or the synthetic division algorithm. The process of dividing polynomials, including cubic polynomials, is an important skill in the context of 5.4 Dividing Polynomials. By understanding the properties of cubic functions and how they can be divided, students can apply these techniques to solve a variety of problems involving polynomial division, which is a fundamental operation in algebra.
• Analyze how the features of cubic functions, such as their roots and points of inflection, can be used to model real-world situations.
  • The distinctive S-shaped curve of cubic functions, along with their potential for up to three real roots and one or more points of inflection, make them well-suited for modeling a wide range of real-world phenomena. For example, the growth of a population over time can be modeled using a cubic function, where the roots represent the points at which the population reaches a stable level, and the points of inflection indicate changes in the rate of growth. Similarly, the trajectory of a projectile, such as a ball or a rocket, can be described by a cubic function, which can be used to predict the object's path and landing location. Understanding the features of cubic functions and how they can be applied to real-world situations is crucial in the context of 5.4 Dividing Polynomials.

Related terms

Polynomial:

A polynomial is an expression that consists of variables and coefficients, where the variables are raised to non-negative integer powers.

Degree:

The degree of a polynomial is the highest exponent of the variable in the expression.

Curve:

A curve is a continuous line that can be described by a mathematical function, often used to represent the shape of an object or the relationship between variables.