key term - Parameter

Definition

A parameter is a variable that is used to define or control the behavior of a mathematical function, system, or process. It is a quantity that can be adjusted or changed to affect the outcome or characteristics of the object or system being studied.

5 Must Know Facts For Your Next Test

1. Parameters are often used in the context of parametric equations, which describe the coordinates of a point as functions of a parameter.
2. In systems of linear equations, parameters can be used to represent unknown quantities that can be varied to find different solutions.
3. The graph of a parametric equation is often referred to as a parametric curve, which can be used to model various shapes and phenomena.
4. Parameters can be used to control the behavior of a function, such as its domain, range, or shape, by adjusting the values of the parameters.
5. The concept of parameters is also important in the study of optimization problems, where parameters are used to define the objective function and constraints.

Review Questions

• Explain how parameters are used in the context of parametric equations.
  • In parametric equations, parameters are used to represent independent variables that can be varied to generate a set of coordinate points. These parameters are typically denoted by a letter, such as $t$, and the coordinates of a point are expressed as functions of the parameter. By varying the parameter, you can trace out a curve or path in the coordinate plane, which is known as a parametric curve. The use of parameters allows for the representation of more complex shapes and phenomena that cannot be easily described using a single equation in terms of $x$ and $y$.
• Describe the role of parameters in systems of linear equations.
  • In a system of linear equations, parameters can be used to represent unknown quantities that can be varied to find different solutions. For example, a system of equations with a parameter $k$ might have the form $2x + 3y = k$ and $4x - y = 5$. By assigning different values to the parameter $k$, you can obtain different solutions for the variables $x$ and $y$. This allows for the exploration of a family of solutions, rather than just a single solution. Parameters in systems of linear equations can be useful in modeling real-world situations where certain quantities may be unknown or variable.
• Analyze how parameters can be used to control the behavior of a function.
  • Parameters in a function can be adjusted to change the function's characteristics, such as its domain, range, or shape. For instance, the function $f(x) = a(x - h)^2 + k$ has three parameters: $a$, $h$, and $k$. By changing the values of these parameters, you can control the function's vertical stretch or compression ($a$), horizontal shift ($h$), and vertical shift ($k$). This allows for the creation of a family of related functions that share a common form but exhibit different behaviors. The ability to manipulate parameters is crucial in many applications, such as modeling real-world phenomena, optimizing systems, and designing complex shapes and curves.

Related terms

Variable:

A quantity that can assume different values within a given context or problem.

Equation:

A mathematical statement that expresses the equality of two expressions.

Function:

A relation between a set of inputs and a set of permissible outputs, where each input is paired with exactly one output.