Negative Functions

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

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

• Graph the function (see Graphing Functions)
  
  
  
  
• The parts of the graph above highlighted in blue are the parts of the graph that are negative (y coordinates are negative). The intervals on the x-axis where the function is negative are highlighted in red. These intervals are the ones we want to find.
  
  
• The red interval on the left goes from negative infinity to the first negative root (see Zoom on right). This interval is (-inf, -2). The second red interval goes from the next negative root to the positive root. This interval is (-1.73205080757, 1.73205080757) (see Finding Roots of a Function).
  

---

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

• Graph the function;
  
  

• The part of the graph above highlighted in blue is the part of the graph that is negative (y coordinates are negative). The interval on the x-axis where the function is negative is highlighted in red. This is the interval we want to find.
  
  
• The red interval goes from the negative root to the positive root. All we need to do is find the roots (see Finding Roots of a Function). The function is negative on the interval (-3/2, 3/2).