CSCA MATHEMATICS · MODULE 02 · FUNCTIONS

Monotonicity of Functions

Increasing and decreasing functions, slope, quadratic turning points, monotonic intervals, and CSCA-style practice.

1. Understanding Monotonicity

Monotonicity describes how the output changes as the input increases. Read graphs from left to right.

Behavior	Output change	Appearance
Increasing	Becomes larger	Graph rises ↗
Decreasing	Becomes smaller	Graph falls ↘
Constant	Remains the same	Graph stays horizontal →

Example 1 · Increasing Function

\[f(x)=2x+1\]

\[f(1)=3,\quad f(2)=5,\quad f(3)=7\]

\[1<2<3\quad\text{and}\quad3<5<7\]

As the input increases, the output increases.

Example 2 · Decreasing Function

\[g(x)=8-2x\]

\[g(1)=6,\quad g(2)=4,\quad g(3)=2\]

\[1<2<3\quad\text{but}\quad6>4>2\]

As the input increases, the output decreases.

Increasing and Decreasing Lines — The positive-slope line rises; the negative-slope line falls. Their common point is not relevant to whether each line increases or decreases.

Important distinction

A few calculated values illustrate a pattern. To prove monotonicity over an entire interval, use the complete graph, the function rule, or the mathematical definition.

2. Definitions of Increasing and Decreasing

Always state the interval on which monotonicity holds. Choose any two inputs in that interval:

\[x_1<x_2\]

2.1 Strictly Increasing

Every larger input produces a strictly larger output.

\[x_1<x_2\quad\Longrightarrow\quad f(x_1)<f(x_2)\]

For instance, if f is strictly increasing on [0,10], then 2 < 7 implies f(2) < f(7).

2.2 Strictly Decreasing

Every larger input produces a strictly smaller output.

\[x_1<x_2\quad\Longrightarrow\quad f(x_1)>f(x_2)\]

For instance, if f is strictly decreasing on [−5,5], then −3 < 4 implies f(−3) > f(4).

Terminology

In this lesson, “increasing” and “decreasing” mean strictly increasing and strictly decreasing unless otherwise specified. Some textbooks use different conventions.

3. Strict versus Non-Strict Monotonicity

3.1 Non-Decreasing

A non-decreasing function may rise or remain flat, but cannot fall.

\[x_1<x_2\quad\Longrightarrow\quad f(x_1)\leq f(x_2)\]

Example 3 · A Flat Section

Let h(x) = 0 when x ≤ 0, and h(x) = x when x > 0.

\[h(-3)=h(-1)=h(0)=0\]

\[h(1)=1,\quad h(2)=2\]

It never decreases, but it is not strictly increasing on its full real domain, because distinct negative inputs have the same output.

3.2 Non-Increasing

A non-increasing function may fall or remain flat, but cannot rise.

\[x_1<x_2\quad\Longrightarrow\quad f(x_1)\geq f(x_2)\]

A constant function such as f(x) = 5 is both non-decreasing and non-increasing; it is not strictly monotonic on any interval containing more than one point.

Type	For x₁ < x₂	Equal outputs allowed?
Strictly increasing	f(x₁) < f(x₂)	No
Non-decreasing	f(x₁) ≤ f(x₂)	Yes
Strictly decreasing	f(x₁) > f(x₂)	No
Non-increasing	f(x₁) ≥ f(x₂)	Yes

4. Identifying Monotonic Intervals from Graphs

Check the domain.
Read the graph from left to right.
Identify rising, falling and horizontal sections.
Locate changes of direction.
Report the corresponding input intervals.

Example 4 · A Graph with Three Sections

Join the points (−4,1), (−1,5), (2,−2), (5,3) in order by straight segments.

Increasing: [−4,−1] and [2,5].

Decreasing: [−1,2].

Direction changes at x = −1 and x = 2.

Piecewise Graph · Increasing, Decreasing, Increasing — The intervals refer to input (horizontal) values, not the vertical range of outputs.

Endpoints

The full closed monotonic sections above include their turning points. Open intervals are also commonly reported in textbooks. Follow the question’s requested endpoint convention.

Important · Increasing Does Not Mean Positive

\[f(x)=x-10\]

\[f(1)=-9,\quad f(3)=-7\]

−9 < −7, so the function increases even though both outputs are negative.

5. Monotonicity of Linear Functions

\[f(x)=mx+b\]

The slope m determines the behavior.

Slope	Behavior	Example
m > 0	Strictly increasing	f(x) = 4x − 7
m < 0	Strictly decreasing	g(x) = 9 − x/2
m = 0	Constant	h(x) = 6

Why Slope Determines Monotonicity

\[f(x_2)-f(x_1)=m(x_2-x_1)\]

Because x₂ − x₁ is positive when x₁ < x₂, the sign of the output difference is the sign of m. The constant b cancels.

Quick Check

\[p(x)=8-3x\]

The slope is −3, so p is strictly decreasing over all real numbers.

6. Monotonicity of Quadratic Functions

\[f(x)=ax^2+bx+c,\quad a
eq0\]

The graph is a parabola. Its vertex is the turning point.

\[h=-\frac{b}{2a},\quad k=f(h)\]

\[f(x)=a(x-h)^2+k\]

6.1 Upward-Opening Quadratic

If a > 0, the function decreases toward the vertex (minimum) and increases afterward.

Example 5 · Minimum at the Vertex

\[f(x)=x^2-4x+3=(x-2)^2-1\]

\[h=2,\quad k=-1\]

Decreasing: (−∞,2]. Increasing: [2,∞). Minimum value: −1 at x = 2.

Upward Quadratic · Vertex and Monotonicity — The minimum occurs at (2, −1). Left of the vertex the function decreases; right of it the function increases.

6.2 Downward-Opening Quadratic

If a < 0, the function increases toward the vertex (maximum) and decreases afterward.

Example 6 · Maximum at the Vertex

\[g(x)=-2(x+1)^2+5\]

\[a=-2,\quad h=-1,\quad k=5\]

Increasing: (−∞,−1]. Decreasing: [−1,∞). Maximum value: 5 at x = −1.

Opening	Increasing on	Decreasing on	Vertex
Upward (a > 0)	[h, ∞)	(−∞, h]	Minimum
Downward (a < 0)	(−∞, h]	[h, ∞)	Maximum

7. Comparing Function Values Using Monotonicity

Example 7 · Increasing

If f is strictly increasing on [0,10], then:

\[3<8\quad\Longrightarrow\quad f(3)<f(8)\]

Example 8 · Decreasing

If g is strictly decreasing on [−5,5], then:

\[-2<4\quad\Longrightarrow\quad g(-2)>g(4)\]

Example 9 · Negative Inputs

f(x) = x² decreases on (−∞,0]. Since −5 < −2:

\[f(-5)>f(-2)\]

\[25>4\]

Check the Interval

If inputs lie on opposite sides of a turning point, a single monotonicity rule may not apply across both. For example, f(−3) = 9 and f(2) = 4 for f(x) = x²; compare directly.

8. Understanding Turning Points

A turning point is where a graph changes from increasing to decreasing (local maximum) or decreasing to increasing (local minimum).

Local Maximum

\[f(x)=-(x-2)^2+7\]

Turning point (2,7); input where the maximum occurs: x = 2; maximum value: 7.

Local Minimum

\[g(x)=(x+3)^2-4\]

Turning point (−3,−4); input where minimum occurs: x = −3; minimum value: −4.

Local extrema compare nearby outputs; absolute extrema compare all outputs in the domain. The vertex of a quadratic on all real numbers is an absolute extremum.

Do not confuse

A zero or x-intercept is not automatically a turning point. Zeros describe where f(x) = 0; turning points describe a change in direction.

9. Restricted Domains

Example 10 · Vertex Inside the Domain

\[f(x)=(x-3)^2+2,\quad 0\leq x\leq5\]

Vertex (3,2). Decreasing on [0,3]; increasing on [3,5]. Do not extend beyond the allowed domain.

Example 11 · Domain Right of the Vertex

\[f(x)=(x-3)^2+2,\quad4\leq x\leq8\]

The whole domain lies to the right of x = 3. Strictly increasing throughout [4,8].

10. Common Mistakes and How to Avoid Them

Mistake	Better approach
Reading the graph right to left	Always follow increasing x.
Assuming negative outputs mean decreasing	Look at output changes, not whether outputs are below zero.
Claiming x² increases everywhere	Locate the vertex: decreases before 0, increases after 0.
Using quadratic roots as monotonic boundaries	Use the vertex rather than x-intercepts.
Comparing inputs from different monotonic sections	Confirm both inputs lie in the same interval of the stated behavior.
Joining separate increasing sections	Report the increasing intervals separately.

11. Practice Quiz

Attempt these ten questions before reading the solutions. These are formative lesson questions, not an official CSCA sample exam.

Question 1 · Definition

For x₁ < x₂, which relationship defines a strictly decreasing function?

Question 2 · Linear function

Classify f(x) = 12 − 5x as increasing, decreasing, or constant, and explain why.

Question 3 · Strict vs. non-strict

A function rises, remains horizontal over a nonzero interval, then rises again. It never falls. Choose: A strictly increasing; B non-decreasing but not strictly increasing; C strictly decreasing; D constant.

Question 4 · Quadratic

Find intervals of increase and decrease for f(x) = x² + 6x + 5.

Question 5 · Downward quadratic

Find vertex and monotonic intervals for g(x) = −(x − 4)² + 9.

Question 6 · Compare outputs

h is strictly decreasing on [−3,8]. Compare h(−1) and h(6).

Question 7 · Negative inputs

For f(x) = x², compare f(−7) and f(−3) using monotonicity.

Question 8 · Turning point

A continuous graph decreases from x = −5 to 1 then increases from 1 to 6; sections meet at (1, −4). Identify the turning point type.

Question 9 · Restricted domain

Find increasing and decreasing intervals of f(x) = (x − 2)² on −1 ≤ x ≤ 4.

Question 10 · Explain the error

A student says: “The graph is below the x-axis, so the function must be decreasing.” Explain with a counterexample.

12. Answers and Explanations

Answer 1

For x₁ < x₂, strictly decreasing means f(x₁) > f(x₂).

Answer 2

Strictly decreasing on all real numbers: slope m = −5 < 0.

Answer 3

B. The function is non-decreasing, but not strictly increasing because the flat portion produces equal outputs.

Answer 4

Vertex x = −3. Decreasing on (−∞,−3] and increasing on [−3,∞).

Answer 5

Vertex (4,9). Increasing on (−∞,4], decreasing on [4,∞).

Answer 6

h(−1) > h(6), because −1 < 6 and h is decreasing throughout the relevant interval.

Answer 7

f(−7) > f(−3); indeed 49 > 9. Both inputs lie left of 0, where x² decreases.

Answer 8

(1,−4) is a local minimum: the graph changes from decreasing to increasing.

Answer 9

Decreasing on [−1,2], increasing on [2,4].

Answer 10

Negative outputs do not determine monotonicity. f(x) = x − 10 on [0,5] has negative outputs but increases because its slope is +1.

13. Lesson Review

Read graphs left to right.
Use the definition to distinguish strictly and non-strictly monotonic functions.
Classify linear functions by slope.
Find the vertex to divide quadratic monotonic intervals.
Compare function values only on a valid monotonic interval.
Distinguish local minima/maxima from roots.
Restrict every answer to the stated domain.

Two Central Comparison Rules