The concepts of domain, limits, derivative, extreme values, monotonicity and concavity have been introduced. Now it is time to combine these concepts together to plot functions and reveal key features of the functions. In order to draw a function, many issues need to be considered. They are domain, intercepts, symmetry, asymptotes, monotonocity, extreme value, concavity and inflection points. Each one is discussed below.

• Domain
  Domain is the range on which the function is defined. Estiblishing or setting the domain is the first step of plotting.
  
• Intercepts
  The intercepts refer to the intersection of the curve and y axis or x axis. In another words, intercepts are the points where the graph cross the axis. By setting x equals 0, the intersection of the curve and the y axis can be found. Theoretically, letting y equals 0, the intersection of the curve and the x axis can be calculated. However, in engineering, complicated functions such as high order polynomial functions might have multiple x intercepts that can make the calculation process difficult. In this case the value of x can be estimated.
  
• Symmetry
  Symmetry can aid in plotting. For some cases only half the curve is determined and reflected about the symmetry axis or point. Although a curve can have symmetry with respect to a line or a point, only symmetry with respect to origin, y axis and periodic symmetry will be introduced here.

• Symmetry about origin
  If f(-x) = -f(x) in the domain of the function, then it is an odd function. The curve of an odd function is symmetry about the origin. This concept can be better understand by finding the symmetry of function f(x) = x³.
  
  First substitute -x into function f(x) = x³ to check the odevity of the function. That gives
  
  f(-x) = (-x)³ = -x³ = -f(x)
  
  Since f(-x) = -f(x) , function f(x) = x³ is an odd function, and thus it is symmetry about the origin. This conclusion is confirmed by its graphic on the left.

• Symmetry about y axis
  If f(-x) = f(x) in the domain of the function, then the function is an even function. The curve of an even function is symmetry about the y axis. This concept can be better understand by finding the symmetry of function f(x) = x².
  First substitute -x into function f(x) = x² to check the odevity of the function. That gives
  
  f(-x) = (-x)² = x² = f(x)
  
  Since f(-x) = -f(x), function f(x) = x² is an even function and is symmetry about the y axis. This conclusion is confirmed by its graphic on the left.

• Periodic symmetry
  If f(x) = f(x + b) where b is a positive constant in the domain of the function, then the function is an periodic function and the smallest b is the period for this function. The curve of a period function repeats every period. For example, function f(x) = sinx is a periodic function with a period of 2π. The entire diagram of the sinx can be plotted by translating the graph of the curve in the range of 0 and 2π as shown on the left.

• Asymptotes
  Basically there are three kinds of asymptotes including horizontal asymptotes, vertical asymptotes, and slant asymptotes. Each one of these will be discussed in the following paragraphs.

• Horizontal asymptotes
  The line y = A is called a horizontal asymptote of the curve y = f(x) if
  
     or    
  
  Although the curve of the function approaches the line y = A, it never actually reaches or crosses.

• Vertical asymptotes
  The line x = A is called a vertical asymptote of the curve y = f(x) if any of the following statements is true:
  
    
  
  
  
  
  
  

Vertical asymptotes IV

• Slant asymptotes
  In the case where the curve approaches a line that is neither horizontal nor vertical, such as the function shown on the left, that line is called an oblique or slant asymptote of the curve. In other words, if
  
  
  
  then the line y = mx + a is called a slant asymptote.

• Monotonocity
  Using the test for monotonicity functions to find the increasing and decreasing interval of the function. If the derivative is larger than zero for all x in (a, b), then the function is increasing on [a, b]. If the derivative is less than zero for all x in (a, b), then the function is decreasing on [a, b].

• Local extreme value
  In engineering and economics, many problems are related to local maximum and minimum values, such as finding the biggest volume and lowest price. Local extreme values are key features of these types of curves. In order to find the local extreme value of a function, the critical point need to be calculated first. Although many methods can be used to separate the local maximum values from the local minimum values, the first derivative test and the second derivative test are the most commonly used methods and are listed below.

• First derivative test
  If the value of the function's derivative changes from positive to negative at x = c, then the function has a local maximum at this point. On the other hand, if the value of the function's derivative changes from negative to positive at x = c, then the function has a local minimum at this point.
  
• Second derivative test
  If at a point the first derivative of the function is 0 and second derivative is larger than 0, then the function has a local minimum at this point. On the other hand, if at a point, the first derivative of the function is 0 and the second derivative is less than 0, then the function has a local maximum at this point.

• Concavity and inflection points
  When a curve is increasing, the bend upward and bend downward curve gives different tendency of a curve. It is similar for a decreasing curve. Computing the second derivative of the function and using the test for concavity can figure out the the bend direction. When the second derivative is larger than 0, the curve is concave downward. On the other hand, when the second derivative is less than 0, the curve is concave upward.

• Sketching
  With the above information, a rough graph can be plotted. First draw the x, y axis and the asymptotes. Second plot the intercepts, extreme values, and the inflection points. Finally, draw the curve passing all these points, increase and decrease according the the test for monotonicity functions, and bend the curve according to the function's concavity.