2.3 Increasing and Decreasing Functions; Average Rate of Change
2.3.1 Learning Objectives
By the end of this section, you should be able to:
- identify intervals on which a function is increasing, decreasing, or constant;
- identify local maximum and minimum values from a graph;
- calculate average rate of change;
- interpret the sign and units of an average rate of change; and
- explain why linear functions have constant rates of change.
2.3.2 Increasing, Decreasing, and Constant Behaviour
A function is increasing on an interval when its outputs rise as the inputs move from left to right. It is decreasing when its outputs fall as the inputs move from left to right. It is constant when its outputs remain unchanged.
More formally, for \(x_1<x_2\):
- \(f(x_1)<f(x_2)\) on an increasing interval;
- \(f(x_1)>f(x_2)\) on a decreasing interval; and
- \(f(x_1)=f(x_2)\) on a constant interval.
Intervals of increase and decrease are reported using input values, so they are intervals on the \(x\)-axis.
Example 2.11 Describing Behaviour from Turning Points
A continuous graph rises until \(x=-2\), falls from \(x=-2\) to \(x=3\), and rises again after \(x=3\). State the intervals on which the function is increasing and decreasing.
Solution
The function is increasing before \(-2\) and after \(3\):
\[ \boxed{(-\infty,-2)\cup(3,\infty)}. \]
It is decreasing between the turning points:
\[ \boxed{(-2,3)}. \]
Exercise 2.11 A graph falls for \(x<-1\), remains constant for \(-1\le x\le2\), and rises for \(x>2\). State the interval on which it is constant.
Show answer
Answer: \(\boxed{[-1,2]}\)
2.3.3 Local Maximum and Minimum Values
A local maximum occurs where a function value is greater than nearby function values. A local minimum occurs where it is less than nearby function values.
The input identifies where the extremum occurs, while the output is the maximum or minimum value. For example, if a graph changes from increasing to decreasing at \((2,7)\), then \(f\) has a local maximum value of \(7\) at \(x=2\).
Example 2.12 Reading Local Extrema
A graph changes from decreasing to increasing at \((-1,-4)\), then changes from increasing to decreasing at \((3,5)\). Identify the local extrema.
Solution
The graph has a local minimum value of \(\boxed{-4}\) at \(x=-1\) and a local maximum value of \(\boxed{5}\) at \(x=3\).
Exercise 2.12 A function changes from increasing to decreasing at \((6,12)\). Identify the local extremum.
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Answer: The function has a local maximum value of \(\boxed{12}\) at \(x=6\).
2.3.4 Average Rate of Change
The average rate of change of \(f\) from \(x=a\) to \(x=b\), where \(a\ne b\), is
\[ \boxed{\frac{f(b)-f(a)}{b-a}}. \]
This quotient compares the change in output with the change in input. Graphically, it is the slope of the secant line through
\[ (a,f(a)) \quad\text{and}\quad (b,f(b)). \]
Example 2.13 Calculating an Average Rate of Change
Find the average rate of change of
\[ f(x)=x^2-3x \]
from \(x=1\) to \(x=5\).
Solution
\[ f(1)=1-3=-2, \qquad f(5)=25-15=10. \]
Therefore,
\[ \frac{f(5)-f(1)}{5-1} =\frac{10-(-2)}4 =\boxed{3}. \]
Exercise 2.13 Find the average rate of change of \(g(x)=2x^2+1\) from \(x=0\) to \(x=3\).
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Answer:
\[ \frac{g(3)-g(0)}{3-0} =\frac{19-1}{3} =\boxed{6}. \]
2.3.5 Interpreting Rate of Change
The sign of an average rate of change describes the overall direction of change:
- a positive rate indicates an overall increase;
- a negative rate indicates an overall decrease; and
- a zero rate indicates equal endpoint outputs.
Units should always be included in applications. If distance is measured in kilometres and time in hours, the average rate of change has units of kilometres per hour.
Example 2.14 Interpreting Average Temperature Change
At 8:00 a.m., the temperature was \(6^\circ\text{C}\). At 2:00 p.m., it was \(18^\circ\text{C}\). Find and interpret the average rate of change.
Solution
Six hours elapsed, so
\[ \frac{18-6}{6}=\boxed{2^\circ\text{C per hour}}. \]
The temperature increased by an average of \(2^\circ\text{C}\) per hour during that interval.
Exercise 2.14 A vehicle’s distance from its starting point changes from \(150\) km at noon to \(90\) km at 2:00 p.m. Find the average rate of change.
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Answer:
\[ \frac{90-150}{2}=\boxed{-30\text{ km/h}}. \]
The negative sign indicates that the vehicle was moving toward the starting point overall.
2.3.6 Linear Functions and Constant Rate of Change
For a linear function
\[ f(x)=mx+b, \]
the average rate of change over every interval equals the slope \(m\). This constant rate is what makes the graph a straight line.
Example 2.15 Verifying a Constant Rate
Find the average rate of change of \(f(x)=-4x+9\) from \(x=1\) to \(x=3\), and from \(x=3\) to \(x=8\).
Solution
From \(1\) to \(3\):
\[ \frac{f(3)-f(1)}{3-1} =\frac{-3-5}{2} =-4. \]
From \(3\) to \(8\):
\[ \frac{f(8)-f(3)}{8-3} =\frac{-23-(-3)}5 =-4. \]
Both rates equal the slope, \(\boxed{-4}\).
2.3.7 Conceptual Takeaways
- Increasing, decreasing, and constant intervals describe how outputs change as inputs move left to right.
- Local extrema compare a function value with nearby values.
- Average rate of change is change in output divided by change in input.
- Graphically, average rate of change is the slope of a secant line.
- The sign and units of a rate are essential to its interpretation.
- Linear functions have the same average rate of change on every interval.
2.3.8 Skills You Should Be Able to Do
- Identify increasing, decreasing, and constant intervals.
- Identify local maximum and minimum values.
- Calculate average rates of change from formulas, tables, or graphs.
- Interpret the signs and units of rates of change.
- Connect average rate of change with secant-line slope.
- Recognize constant rate of change as a defining feature of linear functions.
2.3.9 Practice Problems with Solutions
A graph rises for \(x<-3\), falls for \(-3<x<4\), and rises for \(x>4\). State the increasing intervals.
Show Solution
The graph rises before \(-3\) and after \(4\), so it is increasing on
\[ \boxed{(-\infty,-3)\cup(4,\infty)}. \]
For the graph in Problem 1, state the decreasing interval.
Show Solution
The graph falls between the turning points, so it is decreasing on
\[ \boxed{(-3,4)}. \]
A graph changes from decreasing to increasing at \((2,-5)\). Identify the local extremum.
Show Solution
The change from decreasing to increasing creates a local minimum. Its value is \(\boxed{-5}\), occurring at \(x=2\).
Find the average rate of change of \(f(x)=x^2+2x\) from \(x=1\) to \(x=4\).
Show Solution
\[ f(1)=3, \qquad f(4)=24. \]
\[ \frac{f(4)-f(1)}{4-1} =\frac{24-3}{3} =\boxed{7}. \]
The population of a town increased from \(18,400\) to \(21,100\) over 6 years. Find the average annual rate of change.
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\[ \frac{21,100-18,400}{6} =\frac{2700}{6} =\boxed{450\text{ people per year}}. \]
A water tank contains \(V(t)=900-35t\) litres after \(t\) minutes. Find and interpret the average rate of change from \(t=4\) to \(t=10\).
Show Solution
Because \(V\) is linear, its average rate of change is its slope:
\[ \boxed{-35\text{ L/min}}. \]
The volume decreases by \(35\) litres per minute.
A function has \(f(2)=11\) and \(f(8)=-1\). Find the average rate of change from \(x=2\) to \(x=8\).
Show Solution
\[ \frac{f(8)-f(2)}{8-2} =\frac{-1-11}{6} =\boxed{-2}. \]