A2.6 Distinguish exponential, linear, and quadratic relations

Ontario Grade 11 Mathematics

Mathematical Models

Ontario Grade 11 MBF3C • Study-guide label A2.6

A relation connects an input to an output. For example, a relation might connect the number of weeks to the number of plants in a garden. Some relations change by adding the same amount each time. Others change by adding amounts that themselves change, or by multiplying by the same factor. These patterns help us distinguish linear, quadratic, and exponential relations. In this lesson, you will look for those patterns in words, tables, and equations. You do not need to solve a quadratic equation to identify a quadratic relation.

What you will learn

Describe how linear, quadratic, and exponential relations change.
Use tables, equations, and context to distinguish the three types.
Use first differences, second differences, or ratios when inputs are equally spaced.

1. Start with how the output changes

The input is the value you choose or observe. The output is the value paired with it. In a table, inputs are often shown as

x

and outputs as

y

. A change means the amount an output increases or decreases from one row to the next.

Before comparing changes, check the inputs. The difference tests in this lesson work most directly when the inputs are equally spaced, such as CAD 0, 1, 2, 3. If a table uses uneven input steps, do not compare its output changes as if each step were the same size.

A linear relation has a constant rate of change: for each equal step in the input, the output changes by the same amount. A constant rate can be positive, negative, or zero. For example, adding CAD 4 to a starting amount each week is linear if the same amount is added every week.

A quadratic relation has a changing rate of change. When the inputs are equally spaced, the output's first differences change by a constant amount. Those changes in the first differences are called second differences. This pattern is a useful way to recognize a quadratic relation from a table.

An exponential relation changes by a constant factor over equal input steps. A factor tells how many times as large the next output is. For example, multiplying by

3

at every step gives a constant factor of

3

. The outputs do not have to increase: repeatedly multiplying by a positive factor less than

1

makes them decrease.

\Delta y=y_{\text{next}}-y_{\text{previous}}

Linear: equal input steps give equal output differences.
Quadratic: equal input steps give equal second differences.
Exponential: equal input steps give equal output ratios.

2. Read the pattern in a table

A difference is found by subtracting one output from the next. Keep the order consistent: later output minus earlier output. If the first differences are not constant, compare them in turn. Constant second differences point to a quadratic relation, provided the inputs are equally spaced.

A ratio is found by dividing a later output by the earlier output. Ratios are useful for exponential patterns when the outputs are non-zero and the division is defined. If the same ratio appears between each pair of outputs, the relation is exponential. A table can be rounded, so in real data a pattern may be close to constant rather than exact.

The table below uses inputs that increase by one each time. Its linear outputs rise by

3

each step. Its quadratic outputs have first differences CAD 3, 5, 7 and second differences CAD 2, 2. Its exponential outputs have ratio

2

at every step.

Patterns are evidence, not a reason to ignore context. A repeated addition often describes a fixed fee plus a per-unit charge. Repeated multiplication can describe growth by the same percentage or factor. A quadratic pattern is identified by its changing first differences and constant second differences; do not call a pattern quadratic just because its outputs get larger.

\Delta^2 y=\Delta y_{\text{next}}-\Delta y_{\text{previous}}

Find first differences before second differences.
Use ratios to check for repeated multiplication.
State whether the inputs are equally spaced before using these tests.

3. Connect the patterns to equations and graphs

Equations give another way to recognize the relation. In the equation

y=3x+2

, the output is

2

when the input is

0

, then increases by

3

whenever the input increases by

1

. This is linear. More generally, a linear relation can be written as

y=mx+b

, where

m

is the constant change per one input unit and

b

is the output at input

0

.

A quadratic relation has a squared input, as in

y=x^2

. For equally spaced inputs, its first differences change at a constant rate. The value of the squared input affects how quickly the outputs change. A quadratic relation may also have other terms, such as a term involving

x

, while retaining its quadratic pattern.

An exponential relation has the input in the exponent, as in

y=2^x

. Each increase of

1

in

x

multiplies the output by

2

, so the ratio is constant. By contrast,

y=2x

is linear: it multiplies the input by

2

, and its output differences are constant.

Graphs can support the decision. A linear relation appears as a straight line. A quadratic relation has a curved graph that changes direction or steepness in a different way from a straight line. An exponential graph changes by repeated multiplication and often becomes steeper as it grows. A graph alone may not make a type certain, especially over a small range, so use a table, equation, and context together when available.

y=mx+b,\quad y=x^2,\quad y=2^x

In a linear equation, the input is not squared or placed in an exponent.
In a quadratic equation, a squared input creates the quadratic pattern.
In an exponential equation, the input appears as an exponent.

4. Make a careful choice

Use a consistent routine when a relation is unfamiliar. First read the context and identify the input and output. Next check whether input steps are equal. Then inspect the equation or calculate output differences and ratios. Finally, say which pattern supports your choice.

Do not rely on a single output value or on the fact that the outputs increase. All three types can increase over some interval. The key question is how they change: by a constant amount, by a constant amount in the first differences, or by a constant factor.

For measured data, values may not match a pattern exactly because of rounding or measurement limits. Describe the pattern as approximate when appropriate. The goal here is to distinguish the relation types from their changing patterns, not to claim that every real situation follows a perfect rule.

Identify inputs and outputs before testing a pattern.
Use equal input steps for a fair difference or ratio comparison.
Explain the evidence, not just the relation name.

Three patterns for equally spaced inputs

Type	Outputs	First differences	Second differences or ratios
Linear	2, 5, 8, 11	3, 3, 3	First differences are constant
Quadratic	1, 4, 9, 16	3, 5, 7	Second differences: 2, 2
Exponential	3, 6, 12, 24	3, 6, 12	Ratios: 2, 2, 2

Worked example

Example 1: A table with a constant factor

A culture of plants is counted every week. The table shows the number counted. Distinguish the relation pattern.

Check the input steps
The weeks increase by one each time, so each row represents the same input step. Comparing output changes and ratios is appropriate.
$0,1,2,3$
Compare output differences
Subtract each output from the next. The differences are not equal, so the table does not show a linear pattern.
$6-3=3,\quad 12-6=6,\quad 24-12=12$
Compare output ratios
Divide each later output by the output before it. Each ratio is $2$ , which means the count doubles for every one-week step.
$\frac{6}{3}=2,\quad \frac{12}{6}=2,\quad \frac{24}{12}=2$

Answer: The relation is exponential because equal one-week steps multiply the count by the same factor,

2

.

Check: The outputs match

y=3\cdot 2^x

: at weeks 0,1,2,3, the values are 3,6,12,24.

Worked example

Example 2: Distinguishing a quadratic pattern from a linear one

A table records the height, in centimetres, of an object at equally spaced time readings. The readings are 0,1,2,3,4 seconds, and the heights are 20,27,32,35,36. Distinguish the relation pattern shown by the table.

Find the first differences
The time steps are equal. Subtract consecutive heights to see how much the height changes during each step.
$7,5,3,1$
Check for a linear pattern
The first differences are not equal. Therefore, the height does not change by a constant amount at each equal time step, so this is not linear.
$7\ne 5\ne 3\ne 1$
Find the second differences
Subtract each first difference from the next. The second differences are constant, which identifies a quadratic pattern in this equally spaced table.
$5-7=-2,\quad 3-5=-2,\quad 1-3=-2$

Answer: The table shows a quadratic relation pattern: its first differences are not constant, but its second differences are constant at

-2

.

Check: The pattern is consistent with

y=20+8x-x^2

. Substituting

x=0,1,2,3,4

gives 20,27,32,35,36.

Common mistakes and how to avoid them

Calling any increasing list of outputs linear.

Correction: Check whether the differences are constant. Increasing outputs can also be quadratic or exponential.

Comparing differences when input steps are unequal.

Correction: Check the input spacing first. Differences from unequal steps do not make a fair comparison.

Calling a relation exponential because its outputs get bigger quickly.

Correction: Check whether successive outputs have a constant ratio over equal input steps.

Stopping after finding that first differences are not constant.

Correction: For equally spaced inputs, check the second differences to test for a quadratic pattern.

Lesson summary

Linear relations have constant first differences for equal input steps.
Quadratic relations have constant second differences for equal input steps.
Exponential relations have constant ratios for equal input steps.
Use the context, input spacing, table, and equation as evidence.

Check your understanding

Question 1

For equally spaced inputs, the outputs are CAD 5, 9, 13, 17. Which pattern do they show?

Linear, because the first differences are all $4$
Quadratic, because the outputs increase
Exponential, because the outputs increase
correctIndex: 0,

Show answer and explanation

Linear, because the first differences are all

4

Each consecutive difference is

4

, so the relation has a constant change for equal input steps and is linear.

Question 2

For equally spaced inputs, the outputs are CAD 2, 6, 18, 54. Which pattern do they show?

Linear, because the outputs rise
Quadratic, because the first differences are not equal
Exponential, because each output is $3$ times the previous one
correctIndex:0,

Show answer and explanation

Exponential, because each output is

3

times the previous one

The ratios are

6/2=3

,

18/6=3

, and

54/18=3

. A constant factor of

3

indicates an exponential pattern.

Key terms

Relation: A connection that pairs inputs with outputs.
First difference: The change between consecutive outputs in a table.
Second difference: The change between consecutive first differences.
Ratio: The result of dividing one value by another; here, a later output by the output before it.
Constant: Unchanging; the same value each time.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MBF3C), expectation A2.6. It is a study resource, not an official curriculum publication.

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A2.6 · Distinguish exponential, linear, and quadratic relations