Why doesn't a function crossing from positive to positive guarantee a zero?
Conditions
- The function is continuous.
- The y-values at the endpoints of the interval have the same sign.
Reasoning, step by step
- Observe the y-values for consecutive x-values in the table.
- Check if the sign changes from positive to negative or vice versa.
- If both values are positive (or both negative), note that there is no sign change.
- Conclude that based solely on this information, a zero is not guaranteed in that interval.
Example
The speaker points out that moving from a positive y-value to another positive y-value (like from to ) does not guarantee the graph crosses the x-axis.
Common misconceptions
- Believing that a function must cross the x-axis between any two points, regardless of the signs of the y-values.
- Assuming that if a function is continuous, it must have a zero somewhere in its domain.
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Related questions
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.
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 ) 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.
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