Determination Intervals Of INCREASE or DECREASE

INCREASING FUNCTION DECREASING FUNCTION An increasing function is one that rises as it moves form left to right. Here are some examples: DECREASING FUNCTION A decreasing function is one that falls as it moves form left to right. Here are some examples:

These functions are either increasing or decreasing (one or the other) These functions are either increasing or decreasing (one or the other). They can’t be both at the same time. GREATEST INTEGER SQUARE ROOT LINEAR RATIONAL INCREASING DECREASING

Functions that can be BOTH: There are other functions where part of it is considered to be increasing and part of it decreasing. Functions that can be BOTH: QUADRATIC ABSOLUTE VALUE Both of these functions have something in common – they have a vertex. The vertex is the point which separates the increasing part from the decreasing part. If you look closely at both of these, the part to the left of the vertex is decreasing because it falls as it moves from left to right. The part to the right of the vertex is increasing because it rises as it moves from left to right. It may not always be this way as you can see below. QUADRATIC Increasing Decreasing ABSOLUTE VALUE Increasing Decreasing

If we take the function below, which is a combination of functions joined together, we can see there is more than one section that is increasing and more than one that is decreasing. (-6,-4) (10,-2) (6,2) (3,-4) (0,8) (-9,5) (0,8) Notice that when describing the interval of increase, only the x values from the ordered pairs are used. (6,2) (10,-2) (-6,-4) (3,-4)

If we take the function below, which is a combination of functions joined together, we can see there is more than one section that is increasing and more than one that is decreasing. (-6,-4) (10,-2) (6,2) (3,-4) (0,8) (-6,-4) (10,-2) (6,2) (3,-4) (0,8) (-9,5) Notice again that when describing the interval of decrease, only the x values from the ordered pairs are used. (-9,5)

Intervals of increase: [-3,∞ If you compare the following 2 functions below, one is a quadratic and one is a combination. One is not restricted and the other has a definite beginning and end. x = -6 x = -3 x = 0 x = 6 Because the quadratic is without restriction, it is necessary to use the infinity symbol to describe both intervals of increase and decrease in interval notation. x = 3 The closed points at the extreme left and right of the graph above indicates that there is a restriction and therefore the infinity symbol is not necessary to describe the intervals of increase and decrease for the above function. Intervals of increase: [-3,∞ Intervals of increase: [-6,0] U [3,6] Intervals of decrease: -∞,-3] Intervals of decrease: [-9,-6] U [0,3] U [6,10]

POSITIVE or NEGATIVE Intervals Determination Of POSITIVE or NEGATIVE Intervals

If we take the function below, which is a combination of functions joined together, we can see there is more than one section that is positive and more than one that is negative. (0,8) Positive Interval: (-9,5) Negative Interval: (6,2) The zeros of this function are: (10,-2) (-6,-4) (3,-4)