independent variable

Examples of independent variable in the following topics:

• Graphical Representations of Functions

  • Functions have an independent variable and a dependent variable.
  • When we look at a function such as $f(x)=\frac{1}{2}x$, we call the variable that we are changing, in this case $x$, the independent variable.
  • We assign the value of the function to a variable, in this case $y$, that we call the dependent variable.  
  • We choose a few values for the independent variable, $x$.  
  • Start by choosing values for the independent variable, $x$.

• What is a Quadratic Function?

  • where $a$, $b$, and $c$ are constants and $x$ is the independent variable.  
  • With a quadratic function, pairs of unique independent variables will produce the same dependent variable, with only one exception (the vertex) for a given quadratic function.

• Stretching and Shrinking

  • Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase.
  • If the independent variable $x$ is multiplied by a value less than one, all the x values of the equation will decrease, leading to a "stretched" appearance in the horizontal direction.  

• Graphs of Exponential Functions, Base e

  • The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change (i.e., percentage increase or decrease) in the dependent variable.

• The Cartesian System

  • A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.
  • Therefore, the revenue is the dependent variable ($y$), and the number of cars is the independent variable ($x$).  

• Graphing Equations

  • For an equation with two variables, $x$ and $y$, we need a graph with two axes: an $x$-axis and a $y$-axis.
  • We'll start by choosing a few $x$-values, plugging them into this equation, and solving for the unknown variable $y$.
  • Input values (for the independent variable $x$) from -2 to 2 can be used to obtain output values (the dependent variable $y$) from 5 to 9.  

• Inconsistent and Dependent Systems in Three Variables

  • Systems of equations in three variables are either independent, dependent, or inconsistent; each case can be established algebraically and represented graphically.
  • There are three possible solution scenarios for systems of three equations in three variables:
  • Independent systems have a single solution.
  • The same is true for dependent systems of equations in three variables.
  • Now, notice that we have a system of equations in two variables:

• Inconsistent and Dependent Systems in Two Variables

  • An independent system of equations has exactly one solution $(x,y)$.
  • The previous modules have discussed how to find the solution for an independent system of equations.
  • When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.
  • Systems that are not independent are by definition dependent.
  • A system of equations whose left-hand sides are linearly independent is always consistent.

• Inconsistent and Dependent Systems

  • ) and dependency (are the equations linearly independent?
  • is a system of three equations in the three variables x, y, z.
  • When the equations are independent, each equation contains new information about the variables, and removing any of the equations increases the size of the solution set.
  • For linear equations, logical independence is the same as linear independence.
  • Systems that are not independent are by definition dependent.

• Introduction to Systems of Equations

  • For example, consider the following system of linear equations in two variables:
  • The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
  • is a system of three equations in the three variables $x, y, z$.
  • Each of these possibilities represents a certain type of system of linear equations in two variables.
  • An independent system has exactly one solution pair $(x, y)$.