Increasing and Decreasing Functions 📈📉

Introduction

In calculus, students, one of the most useful ideas is figuring out when a function is going up or down. This helps describe real situations like a growing population, a car speeding up or slowing down, or the height of water in a tank. In this lesson, you will learn what it means for a function to be increasing or decreasing, how to identify those intervals using graphs and derivatives, and why this idea is important in IB Mathematics Analysis and Approaches SL.

Learning objectives

Explain the meaning of increasing and decreasing functions.
Use graphs, tables, and derivatives to identify where a function increases or decreases.
Connect this topic to differentiation, optimization, and kinematics.
Summarize how increasing and decreasing functions fit into calculus.

A key calculus idea is that the derivative tells us about the slope of a function. If the slope is positive, the function tends to rise. If the slope is negative, the function tends to fall. That connection is the heart of this topic.

What do increasing and decreasing mean?

A function is increasing on an interval if, as $x$ gets larger, the values of $f(x)$ also get larger. In simple words, the graph goes up from left to right 📈. A function is decreasing on an interval if, as $x$ gets larger, the values of $f(x)$ get smaller. In simple words, the graph goes down from left to right 📉.

More formally, if $f$ is defined on an interval:

$f$ is increasing when $x_1 < x_2$ implies $f(x_1) < f(x_2)$.
$f$ is decreasing when $x_1 < x_2$ implies $f(x_1) > f(x_2)$.

There is also a weaker version of each idea:

non-decreasing means $f(x_1) f(x_2)$ whenever $x_1 < x_2$.
non-increasing means $f(x_1) f(x_2)$ whenever $x_1 < x_2$.

These ideas are important because a function can rise on one interval and fall on another. Many real-world graphs behave this way. For example, the temperature in a day may rise in the morning, peak at midday, then decrease in the evening 🌞.

Using graphs and tables to spot changes

Before using derivatives, it helps to read a graph carefully. If the graph moves upward as you go from left to right, then the function is increasing. If it moves downward, then it is decreasing. Turning points are especially important because they often mark where the behavior changes.

Consider a simple example:

$$f(x)=x^2$$

This function decreases on the interval $(-,0]$ and increases on the interval $[0,)$. The graph is a U-shape, so values go down until the vertex at $x=0$, then go up after that.

A table of values can also help. Suppose:

$x=-2,-1,0,1,2$
$f(x)=4,1,0,1,4$

The values decrease from $4$ to $0$ and then increase from $0$ to $4$. That tells us the function is decreasing first and increasing later.

In IB questions, you may be asked to describe intervals of increase or decrease from a graph, a table, or a rule. You should always state the interval clearly using correct notation, such as $(-,2)$ or $(1,5]$ depending on the context.

The derivative as a test for increase and decrease

The most powerful calculus tool for this topic is the derivative. The derivative of a function, written as $f'(x)$, gives the gradient of the tangent line to the graph at each point. This gradient tells us whether the function is rising or falling.

The main rule is:

If $f'(x) > 0$ on an interval, then $f(x)$ is increasing on that interval.
If $f'(x) < 0$ on an interval, then $f(x)$ is decreasing on that interval.
If $f'(x) = 0$, the function may be flat at that point, but that alone does not tell us whether the function is increasing or decreasing nearby.

This works because a positive gradient means the graph goes upward as $x$ increases, while a negative gradient means it goes downward.

Example 1

Let $$f(x)=x^3-3x$$

Differentiate:

$$f'(x)=3x^2-3=3(x^2-1)=3(x-1)(x+1)$$

Now find where the derivative is positive or negative.

The critical points occur when $f'(x)=0$, so $x=-1$ and $x=1$.

For $x<-1$, choose $x=-2$: $f'(-2)=9>0$
For $-1<x<1$, choose $x=0$: $f'(0)=-3<0$
For $x>1$, choose $x=2$: $f'(2)=9>0$

So $f(x)$ is increasing on $(-,-1)$ and $(1,)$, and decreasing on $(-1,1)$.

This is a classic example of a graph that rises, falls, then rises again.

Critical points and turning points

A critical point is a point where $f'(x)=0$ or where $f'(x)$ does not exist, provided the function itself is defined there. Critical points are useful because they can mark possible local maximum or minimum points.

A local maximum happens where the function changes from increasing to decreasing. A local minimum happens where the function changes from decreasing to increasing.

However, students, not every point where $f'(x)=0$ is a turning point. For example:

$$f(x)=x^3$$

Then

$$f'(x)=3x^2$$

Since $f'(x) 0$ for all $x 0$ and $f'(x)=0$ only at $x=0$, the function is increasing on both sides of $0$. So $x=0$ is a stationary point, but not a maximum or minimum.

This distinction is important in IB mathematics because questions may ask you to identify whether a stationary point is a turning point. To answer, check the sign of $f'(x)$ on each side of the point.

Interval reasoning and sign charts

A common IB method is to use a sign chart for the derivative. This means you:

Differentiate the function.
Solve $f'(x)=0$.
Split the number line into intervals.
Test the sign of $f'(x)$ in each interval.
Conclude where the function increases or decreases.

Example 2

Let $$g(x)=x^2-4x+1$$

Differentiate:

$$g'(x)=2x-4$$

Set the derivative equal to zero:

$$2x-4=0$$

so $x=2$.

Now test intervals:

If $x<2$, then $g'(x)<0$
If $x>2$, then $g'(x)>0$

So $g(x)$ is decreasing on $(-,2)$ and increasing on $(2,)$.

This also tells us there is a minimum at $x=2$. In fact, completing the square gives:

$$g(x)=(x-2)^2-3$$

which confirms that the minimum value is $-3$.

Sign charts are useful because they organize your reasoning clearly. In exams, this method helps you show logical steps, not just give an answer.

Connection to optimization and kinematics

Increasing and decreasing functions are not just about graphs. They appear in many calculus applications.

Optimization

In optimization, you often want to find the largest or smallest value of a function. For example, a company may want to maximize profit or minimize cost. To solve these problems, you often:

build a function,
differentiate it,
find critical points,
test intervals to see where the function increases or decreases.

A function changing from increasing to decreasing at a point often indicates a maximum. A function changing from decreasing to increasing often indicates a minimum.

Kinematics

In kinematics, if $s(t)$ represents position, then velocity is the derivative:

$$v(t)=s'(t)$$

If $v(t) > 0$, position is increasing, which means the object is moving in the positive direction. If $v(t) < 0$, position is decreasing, which means the object is moving in the negative direction.

For example, if

$$s(t)=t^2-6t$$

then

$$v(t)=2t-6$$

The position decreases when $v(t) < 0$, so when $t<3$, and increases when $t>3$.

This shows how increasing and decreasing functions help describe motion. In real life, this could represent a runner moving back toward a starting point and then moving away again 🏃.

Common mistakes to avoid

Confusing $f(x)$ with $f'(x)$: the derivative tells you about increase or decrease, not the original function directly.
Assuming $f'(x)=0$ means a maximum or minimum: check the sign on both sides.
Forgetting interval notation: always state where the function increases or decreases.
Mixing up rising and falling on a graph: read from left to right.
Ignoring domain restrictions: a function may only be increasing or decreasing on part of its domain.

Conclusion

Increasing and decreasing functions are a central part of calculus because they connect graphs, derivatives, and real-world behavior. students, if you can tell when $f'(x)$ is positive or negative, you can describe how a function behaves, identify turning points, and solve optimization and kinematics problems. This topic builds a strong bridge from differentiation to practical applications in IB Mathematics Analysis and Approaches SL.

Study Notes

A function is increasing when larger $x$-values give larger $f(x)$-values.
A function is decreasing when larger $x$-values give smaller $f(x)$-values.
If $f'(x) > 0$, the function is increasing.
If $f'(x) < 0$, the function is decreasing.
Critical points occur where $f'(x)=0$ or where $f'(x)$ does not exist.
A change from increasing to decreasing gives a local maximum.
A change from decreasing to increasing gives a local minimum.
A stationary point is not always a turning point.
Sign charts are a clear way to study intervals of increase and decrease.
This topic is useful in optimization and kinematics, especially when analyzing motion and extrema.