constant function

Examples of constant function in the following topics:

• Increasing, Decreasing, and Constant Functions

  • Functions can either be constant, increasing as $x$ increases, or decreasing as $x$ increases.
  • In mathematics, a constant function is a function whose values do not vary, regardless of the input into the function.  
  • A function is a constant function if $f(x)=c$ for all values of $x$ and some constant $c$.  
  • Example 1:  Identify the intervals where the function is increasing, decreasing, or constant.
  • Identify whether a function is increasing, decreasing, constant, or none of these

• Introduction to Rational Functions

  • A rational function is one such that $f(x) = \frac{P(x)}{Q(x)}$, where $Q(x) \neq 0$; the domain of a rational function can be calculated.
  • A rational function is any function which can be written as the ratio of two polynomial functions.
  • Any function of one variable, $x$, is called a rational function if, and only if, it can be written in the form:
  • Note that every polynomial function is a rational function with $Q(x) = 1$.
  • A constant function such as $f(x) = \pi$ is a rational function since constants are polynomials.

• Stretching and Shrinking

  • This is accomplished by multiplying either $x$ or $y$ by a constant, respectively.
  • Multiplying the entire function $f(x)$ by a constant greater than one causes all the $y$ values of an equation to increase.
  • where $f(x)$ is some function and $b$ is an arbitrary constant.  
  • Multiplying the independent variable $x$ by a constant greater than one causes all the $x$ values of an equation to increase.
  • where $f(x)$ is some function and $c$ is an arbitrary constant.  

• Translations

  • A translation moves every point in a function a constant distance in a specified direction.
  • To translate a function vertically is to shift the function up or down.
  • where $f(x)$ is some given function and $b$ is the constant that we are adding to cause a translation.
  • To translate a function horizontally is the shift the function left or right.
  • Where $f(x)$ would be the original function, and $a$ is the constant being added or subtracted to cause a horizontal shift.  

• Graphs of Exponential Functions, Base e

  • The function $f(x) = e^x$ is a basic exponential function with some very interesting properties.
  • The basic exponential function, sometimes referred to as the exponential function, is $f(x)=e^{x}$, where $e$ is the number (approximately 2.718281828) described previously.
  • $y=e^x$ is the only function with this property.
  • 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.
  • Also, because the the growth rate of a population of bacteria in a petri dish is proportional to its size, the number of bacteria in the dish at a given time can be modeled by an exponential function such as $y=Ae^{kt}$ where $A$ is the number of bacteria present initially (at time $t=0$) and $k$ is a constant called the growth constant.

• What is a Quadratic Function?

  • where $a$, $b$, and $c$ are constants and $x$ is the independent variable.  
  • The constants $b$and $c$ can take any finite value, and $a$ can take any finite value other than $0$.
  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • When all constants are known, a quadratic equation can be solved as to find a solution of $x$.  
  • With a linear function, each input has an individual, unique output (assuming the output is not a constant).  

• What is a Linear Function?

  • A linear function is an algebraic equation in which each term is either a constant or the product of a constant and (the first power of) a single variable.
  • For example, a common equation, $y=mx+b$, (namely the slope-intercept form, which we will learn more about later) is a linear function because it meets both criteria with $x$ and $y$ as variables and $m$ and $b$ as constants.  
  • In the linear function graphs below, the constant, $m$, determines the slope or gradient of that line, and the constant term, $b$, determines the point at which the line crosses the $y$-axis, otherwise known as the $y$-intercept.
  • The blue line, $y=\frac{1}{2}x-3$ and the red line, $y=-x+5$ are both linear functions.  
  • Identify what makes a function linear and the characteristics of a linear function

• Natural Logarithms

  • The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828.
  • Just as the exponential function with base $e$ arises naturally in many calculus contexts, the natural logarithm, which is the inverse function of the exponential with base $e$ also arises in naturally in many contexts.
  • It is used much more frequently in physics, chemistry, and higher mathematics than other logarithmic functions.
  • For example, the doubling time for a population which is growing exponentially is usually given as ${\ln 2 \over k}$ where $k$ is the growth rate, and the half-life of a radioactive substance is usually given as ${\ln 2 \over \lambda}$ where $\lambda$ is the decay constant.
  • The natural logarithm function can be used to solve equations in which the variable is in an exponent.

• Basics of Graphing Polynomial Functions

  • Another easy point to draw is the intersection with the $y$-axis, as this equals the function value in the point zero, which equals the constant term of the polynomial.
  • Conversely, we can easily read the constant term of the polynomial by looking at its intersection with the $y$-axis if its graph is given (and indeed, we can readily read any function value if the graph is given).
  • The graph of a non-constant (univariate) polynomial always tends to infinity when the variable increases indefinitely (in absolute value).
  • Its constant term is between -1 and 0.
  • Its constant term is between 3 and 4.

• Completing the Square

  • The value of $k$ is meant to adjust the function to compensate for the difference between the expanded form of $a(x-h)^2$ and the general quadratic function $ax^2+bx+c$.  
  • However, it is possible to write the original quadratic as the sum of this square and a constant:
  • Thus, the constant $h$ takes the value $-5$ and the constant $k$ takes the value $-3$.
  • Solve for the zeros of a quadratic function by completing the square