Identifying Function Changes and Extrema Through Graph Tracing

TLDR; Identifying increasing/decreasing intervals and relative extrema using graph tracing and coordinate analysis.

⬆️ Increasing and Decreasing Intervals

The process involves finding the intervals for which the function is increasing and decreasing, and determining any relative extrema.

A function is increasing if, as x increases, y increases. Conversely, a function is decreasing if, as x increases, y decreases.

Tracing the graph from left to right allows for the identification of increasing and decreasing intervals: moving uphill indicates the function is increasing, and moving downhill indicates the function is decreasing.

📈 Relative Maximum

Tracing the graph from left to right reveals a high point, which represents a relative maximum.

The coordinates of the relative maximum are found to be (-2, 18). As x increases, the function also increases until this high point is reached.

📉 Relative Minimum

Moving from the high point and coming down the other side reveals a low point, which represents a relative minimum.

The coordinates of the relative minimum are found to be (+2, -14). After passing this low point, the graph starts to move uphill again.

🔄 Function Changes

The function changes from increasing to decreasing at x = -2, and then from decreasing to increasing at x = +2.

This transition is crucial in identifying the intervals of increasing and decreasing function values.

🔢 Interval Determination

The function is increasing from negative infinity to -2, and from 2 to infinity, excluding -2 and +2 respectively.

The function is decreasing on the open interval from -2 to +2, where the relative maximum and minimum occur.

📊 Local Extrema

The high point represents a relative maximum (or local maximum), while the low point represents a relative minimum (or local minimum).

The relative maximum occurs at (-2, 18) and the relative minimum occurs at (+2, -14).

⚠️ Expressing Intervals

It is important to be able to express the intervals using inequalities in addition to interval notation.

For instance, the interval from negative infinity to -2 is expressed as x < -2, and from 2 to infinity as x > 2.