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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.

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 points where a function's graph crosses the x-axis (where $y = 0$) can be referred to as zeros, x-intercepts, roots, or solutions.

Conditions: The function is being viewed as a graph in the coordinate plane.; The context is solving $f(x) = 0$.

The points where a function's graph crosses the x-axis (where $y = 0$) can be referred to as zeros, x-intercepts, roots, or solutions.

Conditions: The function is being viewed as a graph in the coordinate plane.; The context is solving $f(x) = 0$.

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.

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.