Graphs in the xc plane.
Consider the equation
Consider the case when c = - 1 rather than + 1.
Below is the graph of several values of c.
These are the equations of those graphed above. The smaller the value of c, the less curved the graph. Once c=0 the graph becomes a straight line.
C=1, 2, 3, 4, 15
To see a dynamic graph of different c values, click here.
When c is positive the graph of the equation looks like the graph below which is the graph of
When we consider graphing 2x+b=0 in the bx plane, we see that it has an interesting relationship with the solutions in the xy-plane. That is to say that 2x+b=0 intersects
where the solutions in the xy-plane are located.
For example, if you click here and go to the graph, you can note that 2x+b=0 intersects
at two points. These two points are (0.697, -1.394) and
Using the quadratic formula to solve in the xy-plane we get,
or x = 4.303 and .697
The solutions for the quadratic equations can easily be seen on the graph. When graphing
the solution is x=0.
However, when there are no real solutions, the graph indicates that the solution is not satisfied in the region shown. Click here to compare the graph of an equation with no real solutions and one with two real solutions.