Increasing Functions

The graph of a function is all we need to determine where it is increasing.

Find the intervals on the x axis over which the polynomial funtion f(x) = x³ + 2x² - 3x - 6 is increasing.

• Graph the function (see Graphing Functions)
  
  

• The parts of the graph above highlighted in blue are the parts of the graph that are increasing (y coordinates are increasing as the x coordinates increase). The intervals on the x-axis where the function is increasing is highlighted in red. These intervals are the ones we want to find.
  
  
• The red interval on the left goes from minus infinity to the x-coordinate of the point with the relative maximum value. We find this to be x= (see Finding Relative Maximum Values). The red interval on the right goes from the x-coordinate of the relative minimum point to plus infinity. We find this to be x= (see Finding Relative Minimum Values).
  

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Over what intervals is the rational funtion f(x) = increasing?

• Graph the function;
  
  

• The part of the graph above highlighted in blue is the part of the graph that is increasing (y coordinates are increasing as the x coordinates increase). The interval on the x-axis where the function is increasing is highlighted in red. This is the interval we want to find.
  
  
• The red interval goes from the x-coordinate of the relative minimum point to plus infinity. We find this to be x= (see Finding Relative Minimum Values).