Why does a change from positive to negative imply a zero for a continuous function?

To find an interval that must contain a zero, inspect the signs of the y-values in the table for consecutive x-values. If the sign changes from positive to negative or vice versa between two x-values, the Intermediate Value Theorem guarantees a zero in that interval for a continuous function.

Conditions: The function is treated as continuous.; The table contains discrete sample points of the function.

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.

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.