For a given function f(x), real zeros are determined in the interval [xS,xE].
Additionally, the function is graphed for this interval.
To find the zeros, first, intervals with sign changes are identified within the given interval.
In such an interval, a function that is continuous in that interval has at least one zero, according to the Intermediate Value Theorem.
In each of these identified intervals, a zero is then determined using Newton's method.
Alternatively, the Bisection Method or the Regula Falsi Method can also be used.
To also locate zeros that are points of tangency with the x-axis,
a search is also conducted for possible zeros near local extrema.
If the intersection points of two functions given as y=g(x) and y=h(x) are to be determined,
this can be reduced to finding the zeros of a function.
From y=g(x) and y=h(x), by taking the difference of the two equations, it follows:
g(x) - h(x) = 0.
The solutions to this equation are then the points of intersection of g(x) and h(x)..
Example:
The points of intersection of the line y = 5x + 4 with the parabola y = x2 - 2 are to be determined.
Then the zeros of the function
5x + 4 - (x2 - 2)
are to be found or zeros of
-x2 + 5x + 6