Nonlinear Relationship Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Nonlinear Relationship.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A relationship between two quantities where the rate of change is not constant—the graph is curved, not a straight line.

Not a straight line—it curves. Compound interest grows faster and faster.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A nonlinear relationship has no single steady rate, so its graph bends instead of staying straight.

Common stuck point: The procedure for nonlinear relationship is the easy part; the trap is fitting a straight line to two points and declaring it linear. Asking "Do equal steps in $x$ give changing (not constant) changes in $y$ ?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do equal steps in $x$ give changing (not constant) changes in $y$ ?

Worked Examples

Example 1

medium

The area of a square is

A = x^2

. Compare how

A

changes when

x

goes from 1 to 2, then 2 to 3. Is this relationship linear?

Answer

Not linear — the rate of change increases

First step

A(1)=1, A(2)=4, A(3)=9

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Example 2

hard

For

f(x) = x^2 - 4x + 3

, find the vertex and determine whether the parabola opens up or down.

Example 3

medium

For

y = x^2 + 3

, compute the rate of change from

x=0

x=1

, and from

x=2

x=3

Example 4

medium

For

f(x) = x^2 - 6x + 5

, find the vertex and the

x

-intercepts.

Example 5

hard

Two plans: Plan A pays $20 + $5/hour; Plan B pays $2 per hour worked squared ($2 h²). Find for which

h

Plan B first exceeds Plan A.

Example 6

challenge

Average velocity of a falling object is

v_\text{avg} = (v_0 + v_f)/2

. If

v = 9.8t

(from rest), find

v_\text{avg}

over

t \in [0, 4]

and use it to find distance.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

medium

For

y = x^2

, calculate values at

x = -2, -1, 0, 1, 2

. Is the graph symmetric?

Example 2

hard

Determine whether

y = 2^x

is linear or nonlinear by examining the ratio

y_{n+1}/y_n

for

x = 0, 1, 2, 3

Example 3

easy

y = x^2

linear or nonlinear?

Example 4

easy

y = 3x + 2

linear or nonlinear?

Example 5

easy

A table's

y

-values go

2, 4, 8, 16

x = 1,2,3,4

. Linear or nonlinear?

Example 6

easy

Is the graph of

y = x^3

a straight line?

Example 7

easy

Linear or nonlinear:

y

increases by

3

each time

x

increases by

1

Example 8

easy

y = \frac{1}{x}

linear?

Example 9

easy

Does a curve that bends have a constant rate of change?

Example 10

easy

Linear or nonlinear: area of a square as a function of its side,

A = s^2

Example 11

medium

Table

(x,y)=(1,1),(2,4),(3,9),(4,16)

. Linear or nonlinear? Identify the rule.

Example 12

medium

Table

(x,y)=(0,1),(1,2),(2,4),(3,8)

. Is the change a constant difference or constant ratio?

Example 13

medium

For

y = x^2

, find the rate of change from

x=1

x=2

and from

x=2

x=3

. Are they equal?

Example 14

medium

Is the perimeter of a square,

P = 4s

, linear or nonlinear in

s

Example 15

medium

Data:

(1,3),(2,5),(3,7),(4,9)

. Linear or nonlinear? Give the equation if linear.

Example 16

medium

Compound interest doubles money every period:

\$100, \$200, \$400, \$800

. Linear or nonlinear?

Example 17

medium

Does

y = 2^x

ever have a constant rate of change? Check

x=0\to1

and

x=1\to2

Example 18

challenge

Data

(x,y)=(1,2),(2,6),(3,12),(4,20)

. Show it is nonlinear and find a rule.

Example 19

challenge

Two plans: A grows by \$5/week from \$0; B doubles weekly from \$1. After how many weeks does B first exceed A? Show both are different relationship types.

Example 20

challenge

Explain why

y = x^2

has constant second differences but not constant first differences, using

x = 1,2,3,4

Example 21

medium

Data

(x,y)=(1,5),(2,5),(3,5),(4,5)

. Linear or nonlinear?

Example 22

medium

Volume of a cube is

V = s^3

. Linear or nonlinear in

s

, and what is

V

s=2

s=4

Example 23

easy

Table:

(x,y) = (0,0), (1,1), (2,4), (3,9)

. Linear or nonlinear?

Example 24

easy

Is the equation

xy = 12

linear or nonlinear?

Example 25

easy

Data:

(x,y) = (1,5),(2,10),(3,15),(4,20)

. Linear or nonlinear? Give the equation.

Example 26

medium

Table

(x,y)=(0,1),(1,3),(2,9),(3,27)

. Linear or nonlinear? Identify the rule.

Example 27

medium

Determine whether

(x,y) = (1,2),(2,4),(3,6),(4,8)

is linear and find its slope and intercept.

Example 28

medium

For

y = x^2

, find the first differences and second differences for

x = 0, 1, 2, 3, 4

Example 29

medium

Sketch the difference: which grows faster from

x = 0

x = 10

—

y = 2x

(linear) or

y = 2^x

(exponential)?

Example 30

medium

Data:

(x,y) = (1,1), (2,8), (3,27), (4,64)

. Linear or nonlinear? Give the rule.

Example 31

hard

Data

(x,y) = (1,3), (2,9), (3,19), (4,33)

. Show it is nonlinear and find a rule.

Example 32

hard

For

y = 1/x

, calculate

y

x = 1, 2, 4, 8

. Is the rate of change increasing or decreasing?

Example 33

hard

Show that data

(x,y)=(1,2),(2,5),(3,10),(4,17),(5,26)

is nonlinear and find a quadratic rule.

Example 34

challenge

Define

f(x) = a x^2 + b x + c

. Show using finite differences that for equally spaced inputs of step 1, the second differences of

f

are always

2a

← Back to Nonlinear Relationship Formula Explained Practice Mode

Related Concepts

Linear Relationship Exponential Function Quadratic Functions

Background Knowledge

These ideas may be useful before you work through the harder examples.

linear relationship