increasing and decreasing
- When the supply of pineapples goes up does the price go down?
- When medicine dosage goes up does the pain go down?
- When the speed limit goes up does the number of accidents go up?
Our words for this comparison are increase and decrease.
- If the price of pineapples goes down when the supply of pineapples goes
up, then we say the price is decreasing with respect to the supply.
- If the pain goes down when the dosage goes up, then we say that pain
decreases with respect to dosage.
- If the number of accidents goes up when the speed limit goes up, then we say that accidents increases with respect to speed limit.
The function increases on the set , if for EVERY pair of numbers , we have .
When the domain values increase, the function values also increase. When the
domain values decrease, the function values also decrease.
The domain and range change in the same way.
Note: You cannot test the endpoints of an interval to show a function is increasing. It has to increase for EVERY pair of numbers in the interval.
The function decreases on the set , if for EVERY pair of numbers such that , we
have .
When the domain values increase, the function values decrease. When the domain
values decrease, the function values increase.
The domain and range change oppositely.
Note: You cannot test the endpoints of an interval to show a function is
decreasing. It has to decrease for EVERY pair of numbers in the interval.
Graphically, increasing appears as dots to the right being higher than dots to
the left. Decreasing appears as dots on the curve being lower to the right.
Or, the reverse....
Increasing also appears as dots to the left being lower than dots to the right.
Decreasing also appears as dots on the curve being higher to the left.
Let be a function. The graph of is displayed below.
- is increasing decreasing neither on the interval .
- is increasing decreasing neither on the interval .
However, is not decreasing on the interval . To show that a function is not decreasing on an interval, we merely have to give ONE counterexample. That is we need to come up with ONE pair of numbers in the interval where . Let’s select and .
A counterexample is one example where all of the conditions of a statement are
met, yet the conclusion is not true.
One single counterexample shows a statement to be false.
Measuring Rates
Comparing how the connected values in the domain and range change is called a rate. Our symbol for a change in an amount is a Greek Delta, . And, we use fractions to quantify the comparison of changes.
- If the changes in pineapples and price are both positive, then this rate is positive.
- If the changes in pineapples and price are both negative, then this rate is again positive.
- If the changes in pineapples and price are of different sign, one increases while the other decreases, then this rate is negative.
Using rates, we can compare and measure changes in function values over a domain interval.
The rate-of-change of a function across in the domain is given by
This is also referred to as the average rate of change across the interval.
The rate-of-change measures how the function values change relative to changes in the domain.
Let be a function. The graph of is displayed below.
From the graph, we can estimate that over the interval , had a rate-of-change of .
From the graph, we can estimate that over the interval , had a rate-of-change of .
All of this might sound familiar. You have probably seen this when working with slopes of lines, . The rise is the vertical change (change in the value of the function). The run is the horizontal change (change in the domain).
Rate-of-change is a function idea that we will connect up to the geometric idea of slope.
The rate of change over an interval is an overall measurement. It compares the changes occuring over the whole interval. It compares final values to initial values.
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more examples can be found by following this link
More Examples of Visual Behavior