decreasing function

Examples of decreasing function in the following topics:

• Increasing, Decreasing, and Constant Functions

  • Functions can either be constant, increasing as $x$ increases, or decreasing as $x $ increases.
  • Similarly, a function is decreasing on an interval if the function values decrease as the input values increase over that interval.
  • The figure below shows examples of increasing and decreasing intervals on a function.
  • The function is decreasing on on the interval: $ (−2, 2)$.  
  • Identify whether a function is increasing, decreasing, constant, or none of these

• What is a Quadratic Function?

  • A quadratic function is of the general form:
  • A quadratic equation is a specific case of a quadratic function, with the function set equal to zero:
  • Linear functions either always decrease (if they have negative slope) or always increase (if they have positive slope).
  • All quadratic functions both increase and decrease.
  • The slope of a quadratic function, unlike the slope of a linear function, is constantly changing.

• Relative Minima and Maxima

  • While some functions are increasing (or decreasing) over their entire domain, many others are not.  
  • A value of the input where a function changes from increasing to decreasing (as we go from left to right, that is, as the input variable increases) is called a relative maximum.  
  • Similarly, a value of the input where a function changes from decreasing to increasing as the input variable increases is called a relative minimum.
  • A function is also neither increasing nor decreasing at extrema.
  • This line increases towards infinity and decreases towards negative infinity, and has no relative extrema.

• Exponential Decay

  • Just as a variable can increase exponentially as a function of another, it is possible for a variable to decrease exponentially as well.
  • Just as it is possible for a variable to grow exponentially as a function of another, so can the a variable decrease exponentially.
  • Consider the decrease of a population that occurs at a rate proportional to its value.
  • This rate at which the population is decreasing remains constant but as the population is continually decreasing the overall decline becomes less and less steep.
  • After $15$ years there will be $12.5$, the amount by which the substance decreases is itself slowly decreasing.

• The Leading-Term Test

  • Analysis of a polynomial reveals whether the function will increase or decrease as $x$ approaches positive and negative infinity.
  • All polynomial functions of first or higher order either increase or decrease indefinitely as $x$ values grow larger and smaller.
  • Consider the polynomial function:
  • The properties of the leading term and leading coefficient indicate whether $f(x)$ increases or decreases continually as the $x$-values approach positive and negative infinity:
  • If $n$ is even and $a_n$ is positive, the function inclines both to the left and to the right.

• Stretching and Shrinking

  • If the function $f(x)$ is multiplied by a value less than one, all the $y$ values of the equation will decrease, leading to a "shrunken" appearance in the vertical direction.  
  • If $b$ is greater than one the function will undergo vertical stretching, and if $b$ is less than one the function will undergo vertical shrinking.
  • If we want to vertically stretch the function by a factor of three, then the new function becomes:
  • 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.  
  • If $c$ is greater than one the function will undergo horizontal shrinking, and if $c$ is less than one the function will undergo horizontal stretching.

• Tangent as a Function

  • Characteristics of the tangent function can be observed in its graph.
  • The the tangent function can be graphed by plotting points as we did for the sine and cosine functions.
  • Likewise, we can see that $y$ decreases as $x$ approaches $-\frac{\pi}{2}$, because the outputs get smaller and smaller.
  • As with the sine and cosine functions, tangent is a periodic function; its values repeat at regular intervals.
  • The graph of the tangent function is symmetric around the origin, and thus is an odd function.

• 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.
  • For example, because a radioactive substance decays at a rate proportional to the amount of the substance present, the amount of the substance present at a given time can be modeled with an exponential function.

• Graphs of Logarithmic Functions

  • The logarithmic graph begins with a steep climb after $x=0$, but stretches more and more horizontally, its slope ever-decreasing as $x$ increases.
  • The range of the function is all real numbers.
  • At first glance, the graph of the logarithmic function can easily be mistaken for that of the square root function.
  • When graphing without a calculator we use the fact that the inverse of a logarithmic function is an exponential function.
  • The graph of the logarithmic function with base 3 can be generated using the function's inverse.

• Basics of Graphing Exponential Functions

  • The exponential function y=b^x where $b>0$ is a function that will remain proportional to its original value when it grows or decays.
  • At the most basic level, an exponential function is a function in which the variable appears in the exponent.
  • The most basic exponential function is a function of the form $y=b^x$ where $b$ is a positive number.
  • Let us consider the function $y=2^x$.
  • As you connect the points you will notice a smooth curve that crosses the y-axis at the point $(0,1)$ and is decreasing as $x$ takes on larger and larger values.