1. What does monotonicity mean?
Monotonicity describes how a function’s output changes as its input increases.
When reading a graph, move from left to right, following increasing values of \(x\).
- If the graph rises, the function is increasing.
- If the graph falls, the function is decreasing.
- If the graph stays horizontal, the function is constant.
For example, consider:
\[ f(x)=2x+1. \]
As the input increases from \(1\) to \(2\), the output increases:
\[ f(1)=2(1)+1=3, \]\[ f(2)=2(2)+1=5. \]
Therefore:
\[ 1<2 \quad\text{and}\quad f(1)<f(2). \]
This illustrates increasing behavior.
Now consider:
\[ g(x)=7-2x. \]
We obtain:
\[ g(1)=5, \qquad g(2)=3. \]
Here:
\[ 1<2 \quad\text{but}\quad g(1)>g(2). \]
This illustrates decreasing behavior.
Important: A function’s monotonicity must be described on an interval. A function can increase on one interval and decrease on another.
| Behavior | As the input increases | Appearance of the graph |
|---|---|---|
| Increasing | The output increases. | The graph rises. |
| Decreasing | The output decreases. | The graph falls. |
| Constant | The output stays the same. | The graph is horizontal. |
2. Precise definitions
To describe monotonicity precisely, choose any two inputs in the interval under consideration:
\[ x_1<x_2. \]
Then compare their outputs.
2.1 Strictly increasing
A function is strictly increasing on an interval \(I\) if, for every pair of inputs in that interval:
\[ x_1<x_2 \quad\Longrightarrow\quad f(x_1)<f(x_2). \]
In simple words:
A larger input always gives a larger output.
Example:
\[ f(x)=3x-4. \]
If:
\[ x_1<x_2, \]
then:
\[ 3x_1-4<3x_2-4. \]
Therefore, the function is strictly increasing on:
\[ (-\infty,\infty). \]
2.2 Strictly decreasing
A function is strictly decreasing on an interval \(I\) if:
\[ x_1<x_2 \quad\Longrightarrow\quad f(x_1)>f(x_2). \]
In simple words:
A larger input always gives a smaller output.
Example:
\[ g(x)=-2x+5. \]
If:
\[ x_1<x_2, \]
multiplication by a negative number reverses the inequality:
\[ -2x_1>-2x_2. \]
Adding \(5\) gives:
\[ -2x_1+5>-2x_2+5. \]
Therefore:
\[ g(x_1)>g(x_2). \]
The function is strictly decreasing on:
\[ (-\infty,\infty). \]
2.3 Non-decreasing
A function is non-decreasing if:
\[ x_1<x_2 \quad\Longrightarrow\quad f(x_1)\leq f(x_2). \]
The output may rise or stay the same, but it never falls.
For example:
\[ h(x)= \begin{cases} 0, & x\leq 0,\\ x, & x>0. \end{cases} \]
This function is constant for negative inputs and then rises for positive inputs.
It is non-decreasing on its entire domain, but it is not strictly increasing there because different negative inputs give the same output:
\[ -3<-1, \qquad h(-3)=h(-1)=0. \]
2.4 Non-increasing
A function is non-increasing if:
\[ x_1<x_2 \quad\Longrightarrow\quad f(x_1)\geq f(x_2). \]
The output may fall or stay the same, but it never rises.
| Type | Condition when \(x_1<x_2\) | Meaning |
|---|---|---|
| Strictly increasing | \(f(x_1)<f(x_2)\) | Always rises; equal outputs are not allowed. |
| Non-decreasing | \(f(x_1)\leq f(x_2)\) | May rise or stay level; never falls. |
| Strictly decreasing | \(f(x_1)>f(x_2)\) | Always falls; equal outputs are not allowed. |
| Non-increasing | \(f(x_1)\geq f(x_2)\) | May fall or stay level; never rises. |
Terminology for this lesson: Unless otherwise stated, “increasing” means strictly increasing, and “decreasing” means strictly decreasing. Some textbooks use these words differently, so always check their definitions.
A constant function is both non-decreasing and non-increasing. On an interval containing more than one point, it is neither strictly increasing nor strictly decreasing.
3. Identifying monotonic intervals from a graph
Use this procedure:
- Identify the domain shown.
- Read the graph from left to right.
- Find where it rises, falls, or stays horizontal.
- Identify the input values where its behavior changes.
- Write the corresponding intervals of \(x\).
Worked example 1: A graph with several sections
Suppose a continuous graph:
- Rises from the point \((-4,1)\) to \((-1,5)\).
- Falls from \((-1,5)\) to \((2,-2)\).
- Rises from \((2,-2)\) to \((5,3)\).
Assume each section has the stated behavior throughout.
Using open intervals to report the sections:
\[ \text{Increasing on }(-4,-1)\text{ and }(2,5). \]\[ \text{Decreasing on }(-1,2). \]
The changes occur at:
\[ x=-1 \quad\text{and}\quad x=2. \]
Notice that we report input intervals, such as:
\[ (-1,2), \]
rather than output intervals, such as:
\[ (-2,5). \]
Endpoint convention: We will usually report open intervals when separating graph behavior. An endpoint or turning point can sometimes be included without changing the strict monotonicity of a section. For example, the function in this example is also strictly decreasing on \([-1,2]\).
Increasing does not mean positive
A graph can rise while remaining below the \(x\)-axis.
For example:
\[ f(x)=x-10. \]
Between \(x=1\) and \(x=3\):
\[ f(1)=-9, \qquad f(3)=-7. \]
Because:
\[ -9<-7, \]
the output increases, even though both outputs are negative.
Similarly, a function can decrease while its outputs remain positive.
4. Monotonicity of linear functions
A linear function has the form:
\[ f(x)=mx+b, \]
where \(m\) is the slope.
The sign of \(m\) determines monotonicity:
\[ m>0 \quad\Longrightarrow\quad \text{strictly increasing}. \]\[ m<0 \quad\Longrightarrow\quad \text{strictly decreasing}. \]\[ m=0 \quad\Longrightarrow\quad \text{constant}. \]
The value of \(b\) moves the graph up or down. It does not change its direction.
Worked example 2: Classifying linear functions
a.
\[ f(x)=4x-7. \]
Since:
\[ m=4>0, \]
the function is strictly increasing on:
\[ (-\infty,\infty). \]
b.
\[ g(x)=9-\frac{1}{2}x. \]
Since:
\[ m=-\frac{1}{2}<0, \]
the function is strictly decreasing on:
\[ (-\infty,\infty). \]
c.
\[ h(x)=6. \]
The slope is zero, so the function is constant.
Why the slope rule works
Choose:
\[ x_1<x_2. \]
Then:
\[ f(x_2)-f(x_1) =m(x_2-x_1). \]
Since:
\[ x_2-x_1>0, \]
the sign of the output difference depends entirely on \(m\).
Quick check
Determine the behavior of:
\[ p(x)=8-3x. \]
Answer: It is strictly decreasing because its slope is \(-3\).
5. Monotonicity of quadratic functions
A quadratic function has the form:
\[ f(x)=ax^2+bx+c, \qquad a\neq 0. \]
Its graph is a parabola.
A parabola usually changes direction at its vertex.
The vertex’s horizontal coordinate is:
\[ x_v=-\frac{b}{2a}. \]
Its vertical coordinate is:
\[ y_v=f(x_v). \]
5.1 Upward-opening parabola
When:
\[ a>0, \]
the parabola opens upward.
Reading from left to right:
- The graph decreases until the vertex.
- The graph increases after the vertex.
Thus:
\[ \text{Decreasing on } \left(-\infty,-\frac{b}{2a}\right), \]\[ \text{Increasing on } \left(-\frac{b}{2a},\infty\right). \]
The vertex is a minimum point.
Worked example 3
Find the intervals of increase and decrease for:
\[ f(x)=x^2-4x+3. \]
Step 1: Identify the coefficients.
\[ a=1, \qquad b=-4, \qquad c=3. \]
Step 2: Find the vertex’s horizontal coordinate.
\[ x_v=-\frac{-4}{2(1)}=2. \]
Step 3: Check the opening direction.
Since:
\[ a=1>0, \]
the parabola opens upward.
Therefore:
\[ \boxed{\text{Decreasing on }(-\infty,2)} \]
and:
\[ \boxed{\text{Increasing on }(2,\infty)}. \]
Step 4: Find the minimum point.
\[ f(2)=2^2-4(2)+3=-1. \]
The vertex is:
\[ \boxed{(2,-1)}. \]
Completing the square makes the structure especially clear:
\[ f(x)=(x-2)^2-1. \]
The squared term becomes smaller as \(x\) approaches \(2\) from the left, then larger as \(x\) moves away from \(2\) to the right.
5.2 Downward-opening parabola
When:
\[ a<0, \]
the parabola opens downward.
Reading from left to right:
- The graph increases until the vertex.
- The graph decreases after the vertex.
Thus:
\[ \text{Increasing on } \left(-\infty,-\frac{b}{2a}\right), \]\[ \text{Decreasing on } \left(-\frac{b}{2a},\infty\right). \]
The vertex is a maximum point.
Worked example 4
Find the monotonic intervals of:
\[ g(x)=-2(x+1)^2+5. \]
This is vertex form:
\[ g(x)=a(x-h)^2+k. \]
Here:
\[ a=-2, \qquad h=-1, \qquad k=5. \]
Therefore, the vertex is:
\[ (-1,5). \]
Since the parabola opens downward:
\[ \boxed{\text{Increasing on }(-\infty,-1)} \]
and:
\[ \boxed{\text{Decreasing on }(-1,\infty)}. \]
Its maximum value is:
\[ \boxed{5}, \]
attained at:
\[ x=-1. \]
| Condition | Left of the vertex | Right of the vertex | Vertex |
|---|---|---|---|
| \(a>0\) | Decreasing on \((-\infty,h)\) | Increasing on \((h,\infty)\) | Minimum at \((h,k)\) |
| \(a<0\) | Increasing on \((-\infty,h)\) | Decreasing on \((h,\infty)\) | Maximum at \((h,k)\) |
6. Comparing function values using monotonicity
Monotonicity lets you compare outputs without calculating their exact values.
Worked example 5: Increasing function
Suppose \(f\) is strictly increasing on:
\[ [0,10]. \]
Compare:
\[ f(3) \quad\text{and}\quad f(8). \]
Since:
\[ 3<8, \]
and both inputs belong to the increasing interval:
\[ \boxed{f(3)<f(8)}. \]
Worked example 6: Decreasing function
Suppose \(g\) is strictly decreasing on:
\[ [-5,5]. \]
Compare:
\[ g(-2) \quad\text{and}\quad g(4). \]
Since:
\[ -2<4, \]
decreasing behavior reverses the output order:
\[ \boxed{g(-2)>g(4)}. \]
Worked example 7: Comparing negative inputs of a quadratic
Let:
\[ f(x)=x^2. \]
Compare:
\[ f(-5) \quad\text{and}\quad f(-2). \]
The function is decreasing on:
\[ (-\infty,0]. \]
Since:
\[ -5<-2, \]
we obtain:
\[ \boxed{f(-5)>f(-2)}. \]
Checking:
\[ 25>4. \]
When monotonicity alone is insufficient
For:
\[ f(x)=x^2, \]
the inputs \(-3\) and \(2\) lie on opposite sides of the vertex.
You cannot compare their outputs by saying “the function is increasing” or “the function is decreasing,” because neither description applies across the whole interval from \(-3\) to \(2\).
Instead, calculate:
\[ f(-3)=9, \qquad f(2)=4. \]
Therefore:
\[ f(-3)>f(2). \]
Before using monotonicity, check that both inputs lie in the same interval of the stated behavior.
7. Turning points
A turning point is a point where a graph changes between increasing and decreasing behavior.
Local maximum
If a graph changes from increasing to decreasing, it has a local maximum at the turning point.
The output there is at least as large as nearby outputs.
Example:
\[ f(x)=-(x-2)^2+7. \]
The graph increases before \(x=2\) and decreases after \(x=2\).
The maximum point is:
\[ (2,7). \]
Local minimum
If a graph changes from decreasing to increasing, it has a local minimum at the turning point.
Example:
\[ g(x)=(x+3)^2-4. \]
The graph decreases before \(x=-3\) and increases afterward.
The minimum point is:
\[ (-3,-4). \]
For a quadratic defined on all real numbers, the vertex gives an absolute minimum or maximum as well: its output is the smallest or largest over the entire domain.
For more complicated graphs, a local maximum need not be the highest point of the whole graph.
Do not confuse these quantities:
\[ \text{Turning-point input: }x=2, \]\[ \text{Maximum value: }7, \]\[ \text{Maximum point: }(2,7). \]
A point where the graph crosses the \(x\)-axis is not automatically a turning point.
8. Determining intervals: A complete method
For the functions in this lesson, derivatives are unnecessary.
Method
- Check the domain. Only include inputs where the function is defined.
- Identify the function or graph shape.
- Locate changes of direction. For a quadratic, find its vertex.
- Determine behavior on each section.
- Report intervals using input values.
- Restrict your answer to the given domain.
Worked example 8: A restricted domain
Determine the monotonic intervals of:
\[ f(x)=(x-3)^2+2, \qquad 0\leq x\leq 5. \]
The vertex is:
\[ (3,2). \]
The parabola opens upward, so it decreases before \(x=3\) and increases afterward.
Within the given domain:
\[ \boxed{\text{Strictly decreasing on }[0,3]} \]
and:
\[ \boxed{\text{Strictly increasing on }[3,5]}. \]
Here the endpoints are included because each closed interval satisfies the corresponding strict definition.
Do not give intervals extending to infinity: those inputs are outside the specified domain.
Worked example 9: A flat section
Consider:
\[ p(x)= \begin{cases} x+1, & x<0,\\ 1, & 0\leq x\leq 2,\\ x-1, & x>2. \end{cases} \]
The function:
- Increases before \(x=0\).
- Is constant from \(x=0\) to \(x=2\).
- Increases after \(x=2\).
It is non-decreasing on:
\[ (-\infty,\infty). \]
It is not strictly increasing on its entire domain because:
\[ p(0)=p(1)=p(2)=1. \]
This is the key difference between strictly increasing and non-decreasing.
9. Common mistakes
Mistake 1: Reading from right to left
Monotonicity always describes what happens as \(x\) increases.
Correction: Follow the graph from left to right.
Mistake 2: Using the sign of the output
Negative outputs do not imply decreasing behavior.
Correction: Compare how the outputs change.
Mistake 3: Assuming every quadratic increases everywhere
For:
\[ f(x)=x^2, \]
the function decreases on the negative side and increases on the positive side.
Correction: Split the analysis at the vertex.
Mistake 4: Using the roots as monotonicity boundaries
For:
\[ f(x)=x^2-4x+3, \]
the roots are:
\[ x=1, \qquad x=3. \]
However, the change of direction occurs at:
\[ x=2. \]
Correction: Roots describe where the output is zero. The vertex describes where a quadratic changes direction.
Mistake 5: Treating a few sample values as proof
Several rising values can suggest increasing behavior, but they do not prove that the function increases between those sampled inputs.
Correction: Use the graph’s full behavior, a known function rule, or the definition.
Mistake 6: Combining separate increasing sections into one claim
A function increasing on two separate intervals is not necessarily increasing across their union.
Correction: State each monotonic interval separately unless comparisons across the sections have also been checked.
10. Practice quiz
Try these questions before reading the answers.
Question 1 — Definition
Complete the statement.
A function is strictly decreasing on an interval if:
\[ x_1<x_2 \quad\Longrightarrow\quad \underline{\hspace{3cm}}. \]
Question 2 — Linear function
State whether the function is increasing, decreasing, or constant:
\[ f(x)=12-5x. \]
Question 3 — Strict or non-strict?
A function rises, stays horizontal over an interval, and then rises again. It never falls.
Which description applies to the whole graph?
A. Strictly increasing
B. Non-decreasing, but not strictly increasing
C. Strictly decreasing
D. Constant
Question 4 — Quadratic intervals
Find the intervals of increase and decrease:
\[ f(x)=x^2+6x+5. \]
Question 5 — Downward-opening quadratic
Find the vertex and monotonic intervals:
\[ g(x)=-(x-4)^2+9. \]
Question 6 — Compare outputs
The function \(h\) is strictly decreasing on:
\[ [-3,8]. \]
Compare:
\[ h(-1) \quad\text{and}\quad h(6). \]
Question 7 — Negative inputs
For:
\[ f(x)=x^2, \]
compare:
\[ f(-7) \quad\text{and}\quad f(-3) \]
using monotonicity.
Question 8 — Interpret a graph
A graph decreases for:
\[ -5<x<1 \]
and increases for:
\[ 1<x<6. \]
The sections meet continuously at:
\[ (1,-4). \]
Identify the turning point and its type.
Question 9 — Restricted domain
Determine where the following function increases and decreases:
\[ f(x)=(x-2)^2, \qquad -1\leq x\leq 4. \]
Question 10 — Check a claim
A student says:
“The graph is below the \(x\)-axis, so the function must be decreasing.”
Explain why this is incorrect and give a counterexample.
11. Answers and explanations
Answer 1
\[ \boxed{f(x_1)>f(x_2)}. \]
A larger input gives a smaller output.
Answer 2
\[ \boxed{\text{Strictly decreasing on }(-\infty,\infty)}. \]
The slope is:
\[ m=-5<0. \]
Answer 3
\[ \boxed{\text{B}}. \]
The horizontal section allows equal outputs at different inputs, so strict increase fails.
Answer 4
The vertex’s horizontal coordinate is:
\[ x_v=-\frac{6}{2(1)}=-3. \]
Since the parabola opens upward:
\[ \boxed{\text{Decreasing on }(-\infty,-3)} \]
and:
\[ \boxed{\text{Increasing on }(-3,\infty)}. \]
Answer 5
The vertex is:
\[ \boxed{(4,9)}. \]
The parabola opens downward:
\[ \boxed{\text{Increasing on }(-\infty,4)} \]
and:
\[ \boxed{\text{Decreasing on }(4,\infty)}. \]
Answer 6
Since:
\[ -1<6, \]
and \(h\) is strictly decreasing:
\[ \boxed{h(-1)>h(6)}. \]
Answer 7
The function is decreasing on:
\[ (-\infty,0]. \]
Because:
\[ -7<-3, \]
we obtain:
\[ \boxed{f(-7)>f(-3)}. \]
Answer 8
The graph changes from decreasing to increasing.
Therefore:
\[ \boxed{(1,-4)\text{ is a local minimum point}}. \]
Answer 9
The vertex is at:
\[ x=2. \]
Within the specified domain:
\[ \boxed{\text{Strictly decreasing on }[-1,2]} \]
and:
\[ \boxed{\text{Strictly increasing on }[2,4]}. \]
Answer 10
Being below the \(x\)-axis means the outputs are negative. It does not describe whether they rise or fall.
For example:
\[ f(x)=x-10, \qquad 0\leq x\leq 5. \]
All outputs are negative, but the positive slope means the function is strictly increasing.
12. What you should remember
- Read graphs from left to right.
- Increasing behavior preserves the order of inputs in the outputs.
- Decreasing behavior reverses that order.
- Non-decreasing and non-increasing behavior allow equal outputs.
- A linear function’s slope determines its monotonicity.
- For a quadratic, split the graph at the vertex:
\[ x_v=-\frac{b}{2a}. \]
- Increasing followed by decreasing gives a local maximum.
- Decreasing followed by increasing gives a local minimum.
- Check the domain and monotonic interval before comparing outputs.