Locating roots by change of sign

What you'll learn

1. 1

   Imagine you're walking along a hillside path. When you cross from below sea level to above sea level, you MUST have stepped exactly on sea level at some point! 🏔️

2. 2

   Here's a graph of y = x² - 4. At x = 1, y = -3 (negative). At x = 3, y = 5 (positive). The curve MUST cross y=0 somewhere between x=1 and x=3. 📈

3. 3

   Let's check if f(x) = x³ - 2x - 5 has a root between x = 2 and x = 3.

4. 4

   Test f(x) = x² - 2x - 3 at x = 2 and x = 4. Does the sign change? 🎯

5. 5

   What MUST happen if a continuous function changes sign between two points?

6. 6

   What if f(1) = 4 and f(2) = 4? Does that mean no root between 1 and 2?

7. 7

   If f(1) = -2 and f(4) = 3, what can you conclude?