Intersection Points

By graphing two functions we will be able to find points of intersection.

Find all the points were the graphs of the funtions f(x) = x³ + 2x² - 3x - 6 and g(x) = -x² + 5 intersect.

• Graph the two functions as y1 and y2 (see Graphing Functions)
  
  
  
  
• It is clear that there is at least one point of intersection. By zooming in at the point (-3, -3) we see the graphs have a close encounter, but do not intersect . One method for finding the point of intersection of two graphs is to use the GRAPH/MATH/ISECT option. From the GRAPH menu press the MORE key, select MATH, press the MORE key, and then select ISECT. The calculator will prompts for three pieces of information:
  
  
  • a point on the first curve. This is just to identify one of the two graphs. Just press ENTER.
    
    
  • a point on the second curve. This is just to identify the second graph. Just press ENTER.
    
    
  • a point close to the intersection point. Move the trace cursor using the left or right arrow keys to a point close to the intersection point and press ENTER.
    
    
• The calculator finds the intersection and displays:
  
  Intersection
  x=1.8473221019 y=1.587401052
  
  
• An alternative to using ISECT is to graph f(x) - g(x) and find its roots. These roots will be the x-coordinates of the points of intersection. The y-coordinates can be found by substituting the x-coordinates into either function. The main advantage of this method is that it is easier to locate the roots (we know they will be along the x-axis). Unselect y1 and y2 and define y3 to be y1 - y2. Display y3 (see graph to the right). We find that it has one root (see FINDING ROOTS) at 1.8473221019. For more examples using this method see Solving Equations.