3.8
Decreasing Function
A decreasing function describes a relationship where the output consistently declines as the input increases. This means that for any two input values…
A function is decreasing when its output decreases as the input increases, meaning that as one moves from left to right along the x-axis, the f(x)-values on the graph get smaller.
This behavior is identified by observing whether the graph slopes downward from left to right.
Consider a man running on a track. The time taken and the distance covered for each lap are recorded to determine changes in speed over different intervals.
The average speed—or rate of change—between intervals is determined by calculating the change in distance and dividing it by the change in time between two recorded points.
Next, to identify whether the speed is increasing or decreasing, each lap’s speed is calculated by dividing the distance covered by the time taken for that lap. This helps analyze how the runner’s pace changes from one lap to the next.
When plotted as a speed-versus-time graph, the data shows a consistent decline in speed. This represents a decreasing function, confirming that the runner slows down with each successive lap.
The concept of decreasing functions models various situations where outputs decrease with increasing input, such as battery life or cooling temperature.
Q1: What does it mean for a function to be decreasing?
A decreasing function has outputs that get smaller as inputs increase. Mathematically, for any two input values where x1 < x2, the corresponding outputs satisfy f(x1) > f(x2). On a graph, this appears as a downward slope from left to right, visually confirming the declining relationship between input and output.
Q2: How do you identify a decreasing function on a graph?
A decreasing function is identified by observing whether the graph slopes downward from left to right. As you move along the x-axis from left to right, the f(x)-values consistently get smaller. This visual pattern on a graphs of functions clearly indicates the function is decreasing across that interval.
Q3: What does a negative rate of change indicate about a function?
A negative rate of change indicates a decreasing function. The average rate of change is calculated by dividing the change in output by the change in input. When this ratio is negative across intervals, the function is decreasing, meaning outputs decline as inputs increase.
Q4: How does the derivative help determine if a function is decreasing?
For continuous functions, the derivative f′(x) serves as an indicator of whether a function is decreasing. If f′(x) < 0 for all x in an interval, the function is decreasing on that interval. A negative derivative confirms that outputs are declining as inputs increase.
Q5: What real-world situations can be modeled using decreasing functions?
Decreasing functions model many natural and technological phenomena where outputs decline with increasing input. Examples include the temperature of a cooling object, the voltage of a discharging battery, and the height of a falling object after its peak. These scenarios involve quantities that reduce as time or another input progresses.
Q6: How can you calculate speed changes using the rate of change concept?
Speed changes are calculated by determining the average rate of change between intervals. Divide the change in distance by the change in time between two recorded points to find average speed. Comparing speeds across successive intervals reveals whether the rate is increasing or decreasing over time.
Q7: How does a decreasing function differ from an increasing function?
A decreasing function has outputs that get smaller as inputs increase, while an increasing function has outputs that get larger. On a graph, a decreasing function slopes downward from left to right, whereas an increasing function slopes upward. Understanding both types helps analyze how quantities change across different intervals.