How do you use a table of values to find an interval that must contain a zero?

The Intermediate Value Theorem only guarantees a zero if the function changes sign (from positive to negative or vice versa) over an interval. If both y-values are positive or both are negative, a zero is not guaranteed because the function might stay above or below the axis without crossing it.

Conditions: The function is continuous.; The y-values at the endpoints of the interval have the same sign.

If a continuous function takes a positive value at one x and a negative value at another x, there is at least one zero between those two x-values. This is because the graph must cross the x-axis (where $y=0$) to move from above the axis to below it without breaking.

Conditions: The function is continuous on the interval between the two x-values.; One endpoint value is positive and the other is negative.

The Intermediate Value Theorem states that for a continuous graph, moving from one point to another forces the graph to pass through every y-value lying between the two endpoint y-values. Visually, this means a continuous curve cannot skip intermediate output levels.

Conditions: The graph must be continuous.; The two compared points lie on the same continuous branch of the graph.; The target y-value lies between the y-coordinates of those two points.