Increasing and Decreasing Functions and Relative Extrema
Increasing and Decreasing Functions & Relative Extrema
Increasing and Decreasing Functions
Increasing Function: As x increases, y increases. Tracing the graph from left to right, the graph moves uphill.
Decreasing Function: As x increases, y decreases. Tracing the graph from left to right, the graph moves downhill.
Relative Extrema
Relative Maximum: A high point on the graph.
Example: The coordinates of the relative maximum are (-2, 18). As x increases, the function values also increase until this point.
Relative Minimum: A low point on the graph.
Example: The coordinates of the relative minimum are (2, -14).
Determining Intervals of Increase and Decrease
The function changes from increasing to decreasing at x = -2, and from decreasing to increasing at x = 2.
As we move right on the x-axis, we approach positive infinity (\infty). As we move left, we approach negative infinity (-\infty).
Intervals of Increase and Decrease (Using Interval Notation)
Increasing:
From negative infinity to negative two: (-\infty, -2)
From two to infinity: (2, \infty)
Note: We do not include -2 or 2 in the intervals because the function changes direction at these points.
Decreasing:
From negative two to positive two: (-2, 2)
Local Maximum and Minimum
Local maximum is equal to the y-value at the high point, occuring at x = -2.
The local maximum or relative maximum is 18 when x = -2.
Local minimum is equal to the y-value at the low point, occuring at x = 2.
The local minimum or relative minimum is -14, the function value at the low point at x = 2.
Intervals of Increase and Decrease (Using Inequalities)
Increasing:
x < -2 (equivalent to the interval (-\infty, -2))
x > 2 (equivalent to the interval (2, \infty))
Decreasing:
x > -2 and x < 2 (equivalent to the interval (-2, 2))
Increasing and Decreasing Functions & Relative Extrema
Visualizing Increasing and Decreasing Functions
Increasing Function:
Imagine walking uphill from left to right on the graph.
As you move to the right (x increases), you go up (y increases).
Decreasing Function:
Imagine walking downhill from left to right on the graph.
As you move to the right (x increases), you go down (y decreases).
Relative Extrema (High and Low Points)
Relative Maximum (High Point):
Visualize a peak on the graph.
Example: The peak is at the point (-2, 18).
To the left of this point, you are walking uphill; to the right, you are walking downhill.
Relative Minimum (Low Point):
Visualize a valley or trough on the graph.
Example: The valley is at the point (2, -14).
To the left of this point, you are walking downhill; to the right, you are walking uphill.
Identifying Intervals of Increase and Decrease
The graph changes direction (from uphill to downhill or vice versa) at specific x values.
The function switches from increasing to decreasing at x = -2, and from decreasing to increasing at x = 2.
Think of the x-axis as a road. As you drive to the right, you are heading towards positive infinity (\infty), and as you drive to the left, you are heading towards negative infinity (-\infty).
Expressing Intervals Visually
Increasing (Uphill):
From negative infinity to -2: Imagine driving from the far left (-\infty) until you reach x = -2.
From 2 to infinity: Imagine driving from x = 2 to the far right (\infty).
Note: At x = -2 and x = 2, you are at the turning points, so they are not included in the intervals.
Decreasing (Downhill):
From -2 to 2: Imagine driving from x = -2 to x = 2. This is where you are going downhill.
Understanding Local Maxima and Minima
Local maximum is the height (y-value) of the peak (high point), which occurs at x = -2.
The height of the peak is 18, so the local maximum is 18 when x = -2.
Local minimum is the depth (y-value) of the valley (low point), which occurs at x = 2.
The depth of the valley is -14, so the local minimum is -14 when x = 2.
Using Visual Boundaries (Inequalities)
Increasing (Uphill):
x < -2: Visualize all the points to the left of x = -2, where the graph goes uphill.
x > 2: Visualize all the points to the right of x = 2, where the graph goes uphill.
Decreasing (Downhill):
x > -2 and x < 2: Visualize all the points between x = -2 and x = 2, where the graph goes downhill.