Glossary for Lesson 22: Intermediate Value Theorem

Intermediate Value Theorem

Let’s say that we have a continuous function, f(x).  Then for two points on its graph, (a,f(a)) and (b,f(b)) , that do not have the same y -value (meaning that f(a)\neq f(b) ), then between these two points, the function f will take on every value between f(a) and f(b)

We can use the Intermediate Value Theorem to find real zeros of polynomial functions.  Let’s say that we have the function f(x)=x^3+2 , and we want to find an zero in the interval [-3,1] .  This means that we want to find an x -intercept on the graph of f between the point (-3,-25) to the point (1,3) .  The Intermediate Value Theorem tells us that between the points (-3,-25) and (1,3) , the function will take on every single value between -25 and 30 is between these two numbers, so at some point in this part of the graph, the function will have an x -intercept, since the y -coordinate of any x -intercept is 0.  In other words, because -25 is negative and 3 is positive, the graph has to pass through the x -axis between the two points we’re considering.