Increasing, Decreasing, and Extreme Behavior

The Central Idea

To describe how a function behaves, imagine moving along its graph from left to right while the input \(x\) increases.

• Increasing: the output values rise.
• Decreasing: the output values fall.
• Constant: the output values remain unchanged.

When the direction changes, the graph may form a high point or low point. Those extreme values may be local to one part of the graph or absolute across the entire domain.

Always Read from Left to Right

Increasing and decreasing behavior is described as \(x\) increases. On an ordinary coordinate plane, that means reading the graph from left to right.

A segment that rises visually from left to right is increasing. A segment that falls visually from left to right is decreasing.

Increasing Behavior

A function is increasing on an interval when larger inputs in that interval produce larger outputs.

Formally, for any \(x_1 \[ \boxed{f(x_1) This is the strict definition of increasing used throughout this lesson.

Decreasing Behavior

A function is decreasing on an interval when larger inputs produce smaller outputs.

Formally, for any \(x_1 \[ \boxed{f(x_1)>f(x_2)}. \]

Constant Behavior

A function is constant on an interval when its output does not change as the input changes.

For any \(x_1\) and \(x_2\) in that interval,

Graphically, constant behavior appears as a horizontal segment or horizontal piece of a curve.

Increasing Does Not Mean Positive

The words increasing and positive describe different ideas.

A function can be increasing while all of its outputs are negative. For example, as a graph rises from \((-4,-9)\) to \((-1,-2)\), the outputs are still below the \(x\)-axis, but they are increasing.

Decreasing Does Not Mean Negative

A function can also be decreasing while all of its outputs are positive.

A graph that moves from \((1,10)\) to \((5,3)\) is decreasing because the output falls as \(x\) increases, even though every displayed output is positive.

Behavior Is About Change, Not Location

Positive and negative describe where outputs lie relative to zero. Increasing and decreasing describe how outputs change as the input changes.

Keep these questions separate:

• Is \(f(x)\) above or below zero?
• Is \(f(x)\) rising, falling, or staying constant?

Intervals of Behavior Use Input Values

When asked where a function is increasing or decreasing, report the input interval, not the output values.

If a graph rises as \(x\) moves from \(-3\) to \(2\), then its increasing interval is described using \(-3\) and \(2\).

A Simple Increasing Interval

Suppose a graph rises continuously for

A Simple Decreasing Interval

Suppose the same graph falls continuously for

Why Turning Values Usually Separate the Intervals

At a turning value such as \(x=1\), the graph changes from rising to falling or from falling to rising.

For elementary interval reporting, the turning value is usually used as the boundary between the two behavior intervals:

This keeps the descriptions of the two directional behaviors separate.

Endpoint Conventions for Monotonic Intervals

Textbooks differ on whether finite endpoints are included when reporting an interval of increase or decrease. The underlying behavior does not change.

Malone Math will normally use open intervals when reporting increasing and decreasing behavior, especially at turning values. If a problem or course specifies another convention, follow that convention consistently.

Strictly Increasing and Nondecreasing

Some sources use the word increasing to allow flat portions. To avoid ambiguity, distinguish two ideas.

• Strictly increasing: \(x_1
• Nondecreasing: \(x_1

A nondecreasing function may rise and then remain flat without ever falling.

Strictly Decreasing and Nonincreasing

Likewise,

• Strictly decreasing: \(x_1f(x_2)\).
• Nonincreasing: \(x_1

A nonincreasing function may fall and then remain flat without ever rising.

A Horizontal Plateau

Suppose a graph rises until \(x=2\), remains horizontal from \(x=2\) to \(x=5\), and then falls.

Under the strict terminology used here, the plateau is a constant interval, not part of either the strictly increasing or strictly decreasing interval.

Piecewise Behavior

A graph may change behavior several times. For example, it might be

• decreasing on \(( -\infty,-2)\);
• increasing on \((-2,3)\);
• decreasing again on \((3,\infty)\).

Each change in direction creates a new behavior interval.

Disconnected Graphs Need Separate Intervals

If a graph has a gap, do not automatically combine behavior across the gap.

A function may be increasing on \((-6,-1)\) and also increasing on \((2,5)\), yet the domain contains no values between \(-1\) and \(2\).

Report the intervals separately.

Discrete Graphs Require Care

Increasing and decreasing language applies most naturally to intervals of a continuous or densely defined domain. For a discrete function, compare the outputs in the order of the available inputs.

If

the listed outputs increase as the available inputs increase, even though the graph consists of isolated points.

Increasing Does Not Mean a Straight Line

A curve can increase while its steepness changes continuously. The graph of

is increasing over its entire domain even though it is not a straight line.

Steepness and Direction Are Different

Two graph segments can both be increasing even if one is much steeper than the other.

Increasing tells us the direction of output change. Steepness tells us how rapidly the output changes relative to the input.

A Graph Can Increase More Slowly Without Decreasing

Suppose a curve continues to rise but gradually becomes flatter. Its rate of increase may be shrinking, yet the function is still increasing as long as larger inputs continue to produce larger outputs.

This distinction becomes important later in calculus, where the behavior of a function is separated from the behavior of its rate of change.

Turning Points

A turning point is a point where a graph changes from one directional behavior to the opposite directional behavior.

Common patterns are

and

From Increasing to Decreasing

When a function increases and then decreases, the turning point is a high point relative to nearby graph points.

This pattern signals a local maximum under the usual elementary interpretation.

From Decreasing to Increasing

When a function decreases and then increases, the turning point is a low point relative to nearby graph points.

This pattern signals a local minimum.

Local Maximum

A function has a local maximum at \(x=c\) when \(f(c)\) is at least as large as the function values at nearby inputs in the domain.

In a typical smooth graph, the function rises toward \(x=c\) and falls after \(x=c\).

Local Minimum

A function has a local minimum at \(x=c\) when \(f(c)\) is at least as small as the function values at nearby inputs in the domain.

In a typical smooth graph, the function falls toward \(x=c\) and rises after \(x=c\).

Maximum Point and Maximum Value

Keep the location of an extreme point separate from its output value.

If the graph has a local maximum at

then the local maximum point is \((3,7)\), the maximizing input is \(x=3\), and the local maximum value is

Minimum Point and Minimum Value

If the graph has a local minimum at

then the local minimum point is \((-2,-5)\), the minimizing input is \(x=-2\), and the local minimum value is

Absolute Maximum

An absolute maximum, also called a global maximum, is the greatest output attained anywhere in the domain.

At \(x=c\), this means

Absolute Minimum

An absolute minimum, or global minimum, is the least output attained anywhere in the domain.

At \(x=c\),

Local and Absolute Are Different Comparisons

A local extreme compares one point with nearby points. An absolute extreme compares one point with the entire graph.

A graph can contain several local maxima but only one greatest maximum value, unless several points tie for that same greatest output.

A Local Maximum Need Not Be the Highest Point

Suppose a graph has peaks at \((-2,4)\) and \((5,9)\).

Both can be local maxima. If no point on the graph lies above \(y=9\), then only the second peak attains the absolute maximum value \(9\).

A Local Minimum Need Not Be the Lowest Point

Likewise, valleys at \((1,-3)\) and \((7,-8)\) may both be local minima.

If \(-8\) is the least output anywhere on the graph, then \(-8\) is the absolute minimum value.

An Absolute Extreme Can Also Be Local

An interior point can be both local and absolute.

For

the point \((0,0)\) is a local minimum and also the absolute minimum on \(\mathbb{R}\).

Absolute Extrema Can Occur at Endpoints

If the domain is restricted, an endpoint can produce the greatest or least output on the entire domain.

For example, on \([0,4]\), the function \(f(x)=x\) has absolute minimum value \(0\) at \(x=0\) and absolute maximum value \(4\) at \(x=4\).

Endpoint Local-Extremum Conventions Vary

Whether an endpoint is called a local maximum or minimum depends on the convention being used.

Some courses require a two-sided interior neighborhood and therefore do not count endpoints as local extrema. Others compare only with nearby points in the domain and may count them. Absolute-extremum status is not ambiguous: endpoints can certainly be absolute extrema.

Open Endpoints Are Not Attained

An open circle indicates that the endpoint itself is not part of the graph. Therefore, its output cannot be a maximum or minimum value unless that same output is attained at another included point.

Approaching a Value Is Not the Same as Attaining It

Suppose a graph rises toward the open point \((4,10)\), with outputs getting arbitrarily close to \(10\) but never reaching it.

Then \(10\) is not a maximum value merely because the graph approaches it. A maximum must actually belong to the range.

A Function Can Be Bounded Above Without Having a Maximum

If every output satisfies

then \(10\) can be an upper bound even when no point on the graph has output \(10\). In that case the function may have no absolute maximum.

A Function Can Be Bounded Below Without Having a Minimum

Similarly, if every output satisfies

then \(-3\) can be a lower bound even if the graph never reaches it. The function may then have no absolute minimum.

Unbounded Above

A function is unbounded above if its outputs exceed every proposed upper bound.

Such a function cannot have an absolute maximum.

Unbounded Below

A function is unbounded below if its outputs fall below every proposed lower bound.

Such a function cannot have an absolute minimum.

Extrema and Range Are Closely Connected

If a function has absolute minimum value \(m\) and absolute maximum value \(M\), and every intermediate output occurs, then its range may be

But disconnected graphs and omitted values require more careful range analysis, as developed in the previous lesson.

Repeated Absolute Maximum Values

More than one input can attain the same absolute maximum value.

If

and no output exceeds \(7\), then the function has absolute maximum value \(7\) at both \(x=-2\) and \(x=5\).

Repeated Absolute Minimum Values

The same principle applies to minimum values. A graph may touch the same lowest horizontal level at several inputs.

The minimum value is one output number even though several points may attain it.

Flat Maximum Plateaus

Suppose a graph reaches height \(6\) and remains horizontal from \(x=2\) to \(x=5\), with all other outputs below \(6\).

Then the absolute maximum value is

attained at every input in the plateau interval.

Flat Minimum Plateaus

A horizontal lowest segment behaves similarly. If the graph remains at \(y=-4\) for \(-1\le x\le3\) and lies above that level elsewhere, then the absolute minimum value is \(-4\) throughout that interval.

Local Extrema on Plateaus Need a Convention

Non-strict definitions of local maximum and minimum may classify every point of a flat plateau as a local extreme. Some elementary courses instead reserve the phrase turning point for a sharper change in direction.

When a plateau is present, describe the graph directly and state the interval over which the extreme value is attained.

Corners Can Be Extrema

A graph does not need to be smooth to have a maximum or minimum.

The graph of

has a sharp corner at \((0,0)\), yet that point is both a local and absolute minimum.

Cusps Can Be Extrema

A cusp or other sharp feature can also produce an extreme value. Extrema concern the comparison of output values, not whether the curve has a smooth tangent.

Discontinuities Can Affect Extrema

At a discontinuity, inspect which points are actually included. An open circle is not part of the function, while a filled point at the same input may determine the actual function value.

Example: A Jump with a Local High Point

Suppose a graph approaches an open point at \((2,8)\) from the left, while the actual function value is the filled point \((2,3)\), and nearby values to the right are below \(3\).

The open point \((2,8)\) cannot itself be an extremum because it is not on the graph. Any extreme classification at \(x=2\) must use the included value \(f(2)=3\).

Holes Can Split Monotonic Intervals

A hole can break an otherwise increasing curve into separate domain pieces. Even if the left and right pieces visually line up, an interval containing the missing input is not fully contained in the domain.

Report the increasing behavior on the separate domain intervals.

Vertical Lines Are Not Function Graphs of \(x\)

Increasing and decreasing behavior in this lesson is primarily language for functions \(y=f(x)\).

A vertical line such as \(x=3\) is a relation but not a function of \(x\), so ordinary left-to-right monotonicity does not apply to it in the same way.

The Linear Family

For a nonvertical line

the sign of the slope determines its behavior over its entire domain.

• If \(m>0\), the function is increasing.
• If \(m<0\), the function is decreasing.
• If \(m=0\), the function is constant.

Example: Increasing Line

For

the slope is positive, so the function is increasing on

It has no absolute maximum or minimum because it is unbounded in both vertical directions.

Example: Decreasing Line

For

the slope is negative, so the function is decreasing on

Example: Constant Function

For

every input produces the same output. The function is constant on its entire domain.

Because its range is \(\{5\}\), the value \(5\) is both the absolute maximum and absolute minimum value.

The Basic Upward Parabola

Consider

As \(x\) moves toward \(0\) from the left, the outputs decrease. After \(x=0\), the outputs increase.

Therefore,

The point \((0,0)\) gives the absolute minimum value \(0\).

A Shifted Upward Parabola

For

the vertex is \((3,-4)\). The function is decreasing before \(x=3\) and increasing after \(x=3\):

The absolute minimum value is \(-4\) at \(x=3\).

A Downward Parabola

For

the vertex is \((-2,7)\). The function increases before \(x=-2\) and decreases afterward.

The absolute maximum value is

Absolute Value Behavior

The basic absolute-value function

is decreasing on \((-\infty,0)\) and increasing on \((0,\infty)\). Its corner at \((0,0)\) is the absolute minimum point.

A Cubic Can Be Increasing Everywhere

The function

is strictly increasing over all real numbers:

Although its graph becomes momentarily flat in the tangent sense at the origin, it does not change from increasing to decreasing or vice versa.

A Flat Tangent Is Not Automatically an Extremum

The origin on \(y=x^3\) illustrates an important distinction. A graph may flatten without turning around.

Therefore, a horizontal tangent does not by itself guarantee a local maximum or minimum.

A Turning Point Need Not Have a Horizontal Tangent

The reverse is also true. The corner of \(y=|x|\) is a minimum even though there is no single ordinary tangent slope there.

Graphical extrema are identified first by comparing nearby outputs.

The Reciprocal Function

Consider

Its domain is split at \(x=0\). The function is decreasing on each domain interval:

These intervals cannot be combined because \(x=0\) is not in the domain.

Do Not Combine Across a Domain Break

Saying that \(1/x\) is decreasing on \((-\infty,\infty)\) would be invalid because the interval includes \(x=0\), where the function is undefined.

Behavior intervals must lie entirely inside the domain.

Square-Root Behavior

The basic square-root function

rises throughout its domain. Using the open-interval reporting convention, it is increasing on \((0,\infty)\), while its included endpoint \((0,0)\) gives the absolute minimum value \(0\).

Some sources instead report increasing on \([0,\infty)\). Both statements reflect the same monotonic behavior under different endpoint conventions.

Exponential Growth Behavior

For a standard exponential function

the function is increasing over all real inputs.

Exponential Decay Behavior

For

A Graph May Have No Local Extrema

A function that keeps increasing or keeps decreasing without turning may have no local maxima or minima.

Straight nonhorizontal lines and the cubic \(x^3\) are basic examples.

A Graph May Have Several Local Extrema

Higher-degree polynomials, oscillating functions, and piecewise functions can rise and fall repeatedly.

Each qualifying peak or valley should be identified separately, and the corresponding behavior intervals should be listed in input order.

Extreme Values Need Not Be Integer Coordinates

A graph may turn between grid lines. If the exact coordinates are unknown, estimate them from the graph at the precision supported by the scale.

Do not report false precision such as \(x=2.41736\) from a graph marked only in whole units.

Exact and Estimated Extrema

If an equation or labeled graph gives the exact extreme value, report it exactly. If only a visual graph is available, use an appropriate estimate.

A response such as

is more honest than inventing unsupported exactness.

Graph Scale Affects Behavior Readings

Before identifying turning values, read the axis scales. A visible grid mark may represent \(1\), \(2\), \(10\), \(0.5\), or another interval.

Misreading the scale can shift every reported interval and extreme point.

Graph Window Can Hide Extrema

A viewing window may show only part of a graph. A curve that appears to be increasing throughout the visible region may turn just outside the window.

Make claims about the entire function only when the graph, formula, or context supports them.

Arrows Indicate Continued Behavior

An arrow at the edge of a graph indicates that the graph continues. It does not by itself prove that the same increasing or decreasing trend continues forever unless the drawing or mathematical description clearly indicates that behavior.

Reading Behavior from a Table

A table can also reveal directional behavior. Suppose

The outputs decrease through \(x=0\) and then increase. The table suggests a minimum at \(x=0\) among the listed inputs.

A Table Shows Only the Sampled Inputs

Unless the table represents every allowed input, it does not automatically prove what happens between rows.

A function could rise between two sampled inputs and then return to a lower sampled value. Use the graph or formula when continuous behavior between samples matters.

Reading Behavior from a Formula

Sometimes algebra makes the behavior evident. For example,

is an upward-opening parabola with vertex \((4,1)\). Its graphical behavior can therefore be described without plotting many individual points.

Behavior and Function Families

Familiar function families often carry characteristic behavior patterns:

• positive-slope lines increase everywhere;
• negative-slope lines decrease everywhere;
• upward parabolas decrease then increase;
• downward parabolas increase then decrease;
• basic absolute-value graphs decrease then increase;
• many exponential-growth graphs increase everywhere.

The next lesson develops this family recognition more systematically.

Behavior Under a Vertical Reflection

Reflecting a graph across the \(x\)-axis reverses output order. An increasing segment becomes decreasing, and a local maximum becomes a corresponding local minimum.

The later transformations lesson will formalize these changes.

Behavior Under a Horizontal Shift

Shifting a graph left or right moves the input locations of its turning points without changing the basic rise-and-fall pattern.

For example, \(x^2\) turns at \(x=0\), while \((x-5)^2\) turns at \(x=5\).

Extreme Behavior in Context

In applications, extrema often answer practical questions.

• maximum height of a projectile;
• minimum production cost;
• highest measured temperature;
• lowest inventory level;
• maximum revenue within a modeled interval.

Context Can Restrict the Relevant Domain

A formula may continue mathematically beyond the inputs that make sense in a model. Extreme values should be determined on the relevant domain, not on an unrelated unrestricted extension.

For a projectile model, negative time values may be mathematically defined but physically irrelevant.

Absolute Extrema Depend on the Domain

The same formula can have different absolute extrema when the domain changes.

For \(f(x)=x^2\):

• on \(\mathbb{R}\), it has absolute minimum \(0\) and no maximum;
• on \([-2,3]\), it has absolute minimum \(0\) and absolute maximum \(9\);
• on \((0,3)\), it has neither an attained minimum nor an attained maximum.

Example: Restricted Parabola

Consider

The function decreases on \((-2,0)\) and increases on \((0,3)\).

It has absolute minimum value

and absolute maximum value

Example: Open Restricted Parabola

Now consider

Example: One Local Maximum and One Local Minimum

Suppose a continuous graph is increasing on \((-5,-1)\), decreasing on \((-1,3)\), and increasing on \((3,7)\).

Then the behavior changes

at \(x=-1\), indicating a local maximum there, and changes

at \(x=3\), indicating a local minimum there.

Example: Reading Several Intervals

Suppose a graph is decreasing for \(x<-4\), increasing from \(-4\) to \(1\), constant from \(1\) to \(3\), and decreasing for \(x>3\).

A clear report is

Example: Two Equal Highest Points

Suppose the graph contains peaks at \((-2,6)\) and \((4,6)\), and every other graph point lies below \(y=6\).

The function has one absolute maximum value,

attained at two different inputs, \(x=-2\) and \(x=4\).

Example: Highest Visible Point Is Open

Suppose the highest visible marker is an open circle at \((5,12)\), while all included points have outputs below \(12\) and approach \(12\) without reaching it.

Then \(12\) is not a maximum value. If no smaller output is greatest, the function has no absolute maximum.

Example: Absolute Maximum at an Endpoint

Suppose a graph on \([-3,4]\) rises overall and ends at the filled point \((4,9)\), with no other point above it.

Then

Example: No Absolute Maximum Because the Graph Continues Upward

A graph may contain several visible peaks and still have no absolute maximum if another branch continues upward without bound.

Absolute claims require comparing the entire domain, not only the visually prominent turning points.

Example: Discrete Extreme Values

Consider the discrete relation

Among these included points, the absolute maximum value is

and the absolute minimum value is

Common Error: Reporting \(y\)-Values as Increasing Intervals

If a graph rises from the turning input \(x=-2\) to the turning input \(x=5\), the increasing interval is written with those input values.

Do not use the corresponding heights unless the question asks for extreme values or range.

Common Error: Reversing Left and Right

A graph that looks higher on the left and lower on the right is decreasing, not increasing.

Always imagine \(x\) moving from smaller values to larger values.

Common Error: Confusing Increasing with Above the Axis

A graph below the \(x\)-axis can be increasing, and a graph above the \(x\)-axis can be decreasing.

Direction of change and sign of output are independent properties.

Common Error: Calling Every High Point an Absolute Maximum

A visible peak is not necessarily the highest point on the entire graph.

First determine whether it is local. Then compare its output with every other part of the graph before calling it absolute.

Common Error: Calling an Open Circle an Extremum

An open circle is excluded from the graph. Its coordinates cannot directly supply a maximum or minimum value for the function.

Common Error: Combining Across a Hole or Asymptote

An interval of increase or decrease must remain within the domain. Do not combine behavior across an excluded input simply because the branches have the same visual direction.

Common Error: Treating a Flat Segment as Strictly Increasing

Under the strict definitions in this lesson, a horizontal segment is constant, not increasing or decreasing.

It may still be described as nondecreasing or nonincreasing if those weaker terms are being used.

Common Error: Assuming Every Turning Point Is Smooth

Corners and cusps can also be extrema. Do not require a rounded shape or a horizontal tangent when reading a graph geometrically.

Common Error: Assuming Every Flat Point Is an Extremum

A graph can flatten and continue in the same direction. The function \(x^3\) is the standard example.

An extremum requires a comparison of nearby output values, not merely a flat appearance.

Common Error: Ignoring Endpoints for Absolute Extrema

On a closed restricted domain, the greatest or least output may occur at an endpoint even if the graph never turns there.

Always inspect included endpoints when searching for absolute extrema.

Common Error: Confusing the Extreme Input with the Extreme Value

If the highest point is \((4,11)\), then \(x=4\) is where the maximum occurs, while \(11\) is the maximum value.

State both when the wording of a problem could be ambiguous.

Common Error: Trusting the Window Edge

The top or bottom of a graphing window is not automatically an extremum. The graph may simply continue beyond the displayed region.

A Reliable Procedure for Increasing and Decreasing Intervals

1. Read the horizontal axis and its scale.
2. Identify the domain and any gaps or excluded inputs.
3. Move through the graph from left to right.
4. Mark every input where the directional behavior changes.
5. Separate increasing, decreasing, and constant pieces.
6. Use the turning or boundary inputs as interval endpoints.
7. Keep intervals separate across holes, jumps, or asymptotes.
8. Report intervals using the requested endpoint convention.
9. Verify that the intervals use \(x\)-values rather than output values.

A Reliable Procedure for Finding Local Extrema

1. Locate places where the graph changes directional behavior.
2. Check whether increasing changes to decreasing.
3. Classify that point as a local maximum when appropriate.
4. Check whether decreasing changes to increasing.
5. Classify that point as a local minimum when appropriate.
6. Inspect corners, cusps, jumps, and plateaus as well as smooth turns.
7. Use only included graph points.
8. State the input location, the point, or the extreme value as requested.

A Reliable Procedure for Finding Absolute Extrema

1. Determine the full domain under consideration.
2. Identify all included endpoints.
3. Identify all local high and low points.
4. Inspect isolated points and discontinuities.
5. Determine whether the graph is unbounded above or below.
6. Compare all candidate output values across the entire domain.
7. The greatest attained output is the absolute maximum value.
8. The least attained output is the absolute minimum value.
9. If no greatest or least output is attained, state that the corresponding absolute extremum does not exist.

A Fast Visual Scan

For a first pass, trace the graph from left to right and say quietly:

up, down, flat, up, ...

Then mark the input values where those words change. Those values organize the monotonic intervals and often identify candidate extrema.

A Fast Check for an Absolute Maximum

Sweep a horizontal line downward from above the graph. The first included graph point or points it reaches, if such a highest level exists, attain the absolute maximum value.

A Fast Check for an Absolute Minimum

Sweep a horizontal line upward from below the graph. The first included point or points it reaches, if such a lowest level exists, attain the absolute minimum value.

Graphical Analysis Before Calculus

Everything in this lesson can be understood from graphs and basic function ideas alone.

Later, calculus will provide algebraic tools for locating many of the same intervals and extreme points using derivatives. The graphical meaning comes first: rise, fall, flatten, and turn.

Derivative Preview

When derivatives are introduced later, positive derivative values will generally correspond to increasing behavior, negative derivative values to decreasing behavior, and many interior extrema will occur at inputs where a derivative is zero or undefined.

This is a preview only. The graph remains the primary object in this lesson.

Practice 1: Basic Rise and Fall

A continuous graph is decreasing for \(x<-2\) and increasing for \(x>-2\). State the intervals of decrease and increase, and identify the type of extreme behavior at \(x=-2\).

Practice 2: Upward Parabola

For

state the increasing and decreasing intervals and identify the absolute minimum value.

Practice 3: Downward Parabola

For

state the increasing and decreasing intervals and identify the absolute maximum value.

Practice 4: Several Behaviors

A graph is increasing on \((-6,-1)\), constant on \((-1,2)\), and decreasing on \((2,7)\). Describe the behavior in words as \(x\) moves from left to right.

Practice 5: Local versus Absolute

A graph has local maxima at \((-3,4)\) and \((5,9)\), and no point lies above \(y=9\). Which point or points are local maxima, and what is the absolute maximum value?

Practice 6: Open Endpoint

A graph on \(0

Practice 7: Restricted Line

Consider

State its monotonic behavior and find its absolute minimum and maximum values.

Practice 8: Domain Break

The function

is decreasing on each side of its excluded input. State the two decreasing intervals.

Practice 9: Discrete Extrema

Find the absolute maximum and minimum values of

Practice 10: Identify the Error

A student sees a graph with a local maximum at \((2,7)\) and says, “The function is increasing on \((-1,7)\) because the \(y\)-values rise from \(-1\) to \(7\).” Explain the conceptual error.

Practice Answers

Practice 1

The graph decreases before \(x=-2\) and increases afterward:

The change from decreasing to increasing indicates a local minimum at \(x=-2\).

Practice 2

The vertex is \((4,-9)\). Therefore,

The absolute minimum value is

Practice 3

The vertex is \((-1,6)\). Therefore,

The absolute maximum value is

Practice 4

Reading left to right, the graph rises, then remains flat, then falls.

Practice 5

Both \((-3,4)\) and \((5,9)\) are local maximum points. Because no graph point lies above \(y=9\), the absolute maximum value is

Practice 6

No. The outputs approach \(12\), but the open endpoint means \(12\) is not attained there. Since the function is strictly increasing on \((0,5)\), no smaller attained output can be the greatest output. Therefore,

Practice 7

The slope is positive, so the function is increasing across the restricted domain. Evaluate the included endpoints:

Thus,

Practice 8

Because \(x=0\) is excluded, the decreasing behavior occurs on two separate intervals:

Practice 9

Among the listed outputs \(3,8,-2,5\), the greatest is \(8\) and the least is \(-2\). Therefore,

Practice 10

Increasing intervals are reported using input values, not the low and high output values. The coordinate \(7\) is the output at the local maximum, not automatically an endpoint of an increasing interval. The student must identify the \(x\)-values over which the graph rises.

What to Remember

Increasing, decreasing, and extreme behavior describe how a function changes as its inputs move from left to right.

• increasing means larger inputs produce larger outputs;
• decreasing means larger inputs produce smaller outputs;
• constant behavior appears as a horizontal graph segment;
• behavior intervals are reported with \(x\)-values;
• domain breaks split behavior intervals;
• increasing does not mean positive, and decreasing does not mean negative;
• local extrema compare nearby outputs;
• absolute extrema compare the entire domain;
• maximum and minimum values must actually be attained;
• open circles cannot themselves supply attained extrema;
• included endpoints can be absolute extrema;
• corners and cusps can be extrema even without smooth tangents;
• a flat tangent does not automatically create an extremum;
• plateaus require careful wording and convention awareness;
• graph scale and graph window must be checked before making claims.

The core directional tests are

Where We Go Next

Increasing and decreasing intervals describe how a graph moves, while local and absolute extrema identify important high and low values. Another layer of graphical structure comes from recognizing repeated shape.

The next lesson, Symmetry and Function Families, examines reflection and origin symmetries and develops recognition of major parent function families from their characteristic graphical forms.