Let’s say that we have a continuous function,. Then for two points on its graph, and , that do not have the same -value (meaning that ), then between these two points, the function will take on every value between and .
We can use the Intermediate Value Theorem to find real zeros of polynomial functions. Let’s say that we have the function, and we want to find an zero in the interval . This means that we want to find an -intercept on the graph of between the point to the point . The Intermediate Value Theorem tells us that between the points and , the function will take on every single value between and . is between these two numbers, so at some point in this part of the graph, the function will have an -intercept, since the -coordinate of any -intercept is . In other words, because is negative and is positive, the graph has to pass through the -axis between the two points we’re considering.