The general gradient of a graph is given by the gradient function
To find the turning points of a graph we solve the equation
To find the set of values for which the graph is going up or increasing we solve
In general we will get an interval or set of intervals.
To find the set of values for which the graph is going down or decreasing we solve
Again, in general we will get an interval or set of intervals.
Example: Find the set of values of
for which
is increasing.
We solve for
Example: Find the set of values offor which
is decreasing.
We solve for
To see why this is the case we can sketch the graph of
We are looking for those values of x on the graph where the graph is less than 0 ie