Intermediate Value Theorem (IVT)

Graph Polynomials with IVT - How it Works - Video

Intermediate Value Theorem (IVT)

Intermediate Value Theorem (IVT):

Here we have two models. The models are the same except the one on the left is decreasing and the one on the right is increasing.

No matter, which way our graph is going, we will have the same scenario. Remember IVT states there is always a number in between two numbers as long as the two numbers are not equal.

Mathematical that is if f(a) ≠ f(b) for a < b then f takes a value between f(a) and f(b) in the interval [a, b].

Intermediate Value Theorem (IVT) - Model

Intermediate Value Theorem (IVT) - Model:

We already have talked about mathematically. Now, let's talk about it visually. 

If f(a) ≠ f(b) for a < b, then we have some point, Q, between them at y = w. The height of Q is f(c) or w and is in between f(a) and f(b). 

IVT to Find Zeros

IVT to Find Zeros:

We can use this theorem to tells us where the zeros (x-intercepts - roots - answers) are. Remember 0 is neither positive nor negative. If f(a) = positive and f(b) = negative like in the graph on the left, we will have a zero somewhere in between, in this case at z. If f(a) = negative and f(b) = positive like in the graph on the right, we still will have a zero somewhere in between, in this case at z. 

We can use this theorem to graph polynomials.

Example 1

Example 1:

Here we have a chart of the Average Height of Children.

Let's say the average height of a 3 year old is about 38 inches and the average height of 9 year old is about 50 inches. How did you gain those 12 inches? Did you gain the 12 inches in one go? Or did you gain those 12 inches over a period of time?

You gained it over a period of time. At one point, you were 39 inches, 40, inches, ... 48 inches, and 49 inches. The same goes the IVT. There will always be a number in between two numbers if the function is continuous and not equal.

Live Worksheet

Teacher - Edpuzzle Link