B1.6 Distinguish exponential, linear, and quadratic functions

Ontario Grade 11 Mathematics

Exponential Functions

Use equations, tables, and graph shapes to recognize three important patterns

A function gives one output for each input. For example, a function might describe how a quantity changes as time passes. Some quantities increase by the same amount each step. Others increase by a changing amount, or by the same factor. These patterns help identify whether a function is linear, quadratic, or exponential. In this lesson, you will compare their equations, tables, and graphs.

What you will learn

Recognize basic linear, quadratic, and exponential functions from their equations.
Use changes in a table of values to distinguish the three types.
Connect each function type to its typical graph shape and a suitable situation.

1. Prerequisite bridge: inputs, outputs, and changes

An input is the value you choose, often written as

x

. An output is the value the function gives, often written as

y

. A table lists input-output pairs. A graph shows those pairs as points, with the input on the horizontal axis and the output on the vertical axis.

A difference is found by subtracting one value from the next. For instance, if outputs go from

4

to

9

, the change is

5

. A ratio compares values by division. If an output changes from

4

to

12

, the new value is three times the old value, so the ratio is

3

.

When table inputs are evenly spaced, such as CAD 0, 1, 2, 3, compare each output with the one immediately before it. Equal steps between inputs make the pattern of output changes easier to read. We will use first differences, second differences, and ratios as clues.

A function links each input to one output.
A difference measures an amount of change; a ratio measures a multiplicative change.
For table patterns, check that the inputs are equally spaced.

2. The three patterns in plain language

A linear function changes by the same amount for equal steps in the input. For example, a cost that rises by CAD 4 for each item has a constant increase. Its graph is a straight line. In the equation

y=mx+b

,

m

is the rate of change and

b

is the output when the input is zero.

A quadratic function has a squared input as its highest power. Its basic equation is

y=ax^2+bx+c

, where

a

,

b

, and

c

are numbers and

a

is not zero. Its graph is a curve called a parabola. For equally spaced inputs, its first differences do not usually stay constant, but its second differences do.

An exponential function has the input in the exponent. A basic form is

y=ab^x

, where

a

is the starting value and

b

is the factor applied for each increase of one in

x

. Here,

b

is positive and not equal to

1

. For equal input steps, outputs are multiplied by the same factor. The graph usually curves upward when

b>1

and

a>0

, or falls toward zero when

0<b<1

and

a>0

.

The shape alone can sometimes be misleading. A small part of a curve may look almost straight. When an equation or a reliable table is available, use its pattern rather than guessing from appearance.

y=mx+b,\quad y=ax^2+bx+c,\quad y=ab^x

Linear: constant additive change; straight-line graph.
Quadratic: squared input; parabolic graph; constant second differences for equal input steps.
Exponential: input in the exponent; constant multiplicative factor for equal input steps.

3. Multiple representations: read the table, then connect it to symbols

Suppose a table has inputs that increase by one each time. For a linear pattern, subtract consecutive outputs. If the same difference repeats, that is evidence of a linear function. For a quadratic pattern, find the first differences, then compare those differences. If the second differences repeat, that is evidence of a quadratic function.

For an exponential pattern, divide each output by the output before it. If the same ratio repeats, that is evidence of an exponential function. This ratio test works for the basic form when the outputs are nonzero. A zero output or a changed input step can make the test unsuitable as stated.

Equations give another quick clue. A variable raised to the first power in a rule such as

y=3x+2

is linear. A squared variable in a rule such as

y=2x^2-1

signals a quadratic. A variable used as an exponent in a rule such as

y=5(2)^x

signals an exponential function.

The table below shows the patterns for inputs that increase by one. For the quadratic row, the first differences are CAD 3, 5, 7, 9 and the second differences are CAD 2, 2, 2. For the exponential row, each output is twice the one before it.

Use first differences to test for a linear pattern.
Use second differences to test for a quadratic pattern.
Use ratios of consecutive outputs to test for an exponential pattern.

4. Guided example and application

A function rule or data set can be classified by looking for the feature that stays consistent. In the example, each table uses equal steps in its inputs. Check the differences and ratios, then connect the pattern to its likely equation type.

These patterns also suggest what kind of situation a model might describe. A fixed fee plus the same charge for each item can be linear. A quantity that grows by the same percentage over equal time periods can be exponential. A changing quantity whose equal-step differences themselves change by a steady amount can be quadratic. These are clues for choosing a model, not proof that every real situation follows the rule forever.

Classify from a repeated pattern, not from one pair of values.
A real-world context can suggest a function type, but check the data or rule.

5. A reliable decision process

Start with the representation you have. If you have an equation, inspect how the input appears. If you have a table, make sure the inputs are evenly spaced before comparing outputs. If you have a graph, look for a straight line or a curved shape, then use any available equation or values to confirm.

For table data, check first differences first. If they are constant, the pattern is linear. If they are not constant, check second differences. If those are constant, the pattern is quadratic. You can also check ratios when the outputs are nonzero. A repeated ratio points to an exponential pattern.

Do not assume that one test answers every question. A short table may not provide enough information to identify a rule with certainty. State what the evidence shows: for example, that the listed values fit a constant-difference pattern. This is more careful than claiming that a few points prove the function rule.

Read the equation, table, or graph with the same goal: find the defining pattern.
For equally spaced inputs, compare differences and ratios.
Use evidence from more than one representation when it is available.

Patterns for equal steps in the input

Function type	What to compare	Typical clue
Linear	Consecutive output differences	Same difference
Quadratic	Differences, then differences of those differences	Same second difference
Exponential	Consecutive output ratios	Same ratio

Worked example

Classify three patterns from tables

Each table uses inputs that increase by one. Identify the likely function type for each set of outputs: A: CAD 2, 6, 10, 14; B: CAD 1, 4, 9, 16; C: CAD 3, 6, 12, 24.

Test set A
Subtract consecutive outputs. The differences are all $4$ , so the output increases by a constant amount for each unit increase in input. This is the defining pattern of a linear function.
$6-2=4,\quad 10-6=4,\quad 14-10=4$
Test set B
The first differences are not equal. Compare those differences: they increase by $2$ each time. The constant second difference indicates a quadratic pattern for these equally spaced inputs.
$3,\ 5,\ 7\ ;\quad 5-3=2,\quad 7-5=2$
Test set C
Divide each output by the one before it. Each ratio is $2$ , so the outputs double for every unit increase in input. This is the defining pattern of an exponential function.
$\frac{6}{3}=2,\quad \frac{12}{6}=2,\quad \frac{24}{12}=2$

Answer: Set A is linear, set B is quadratic, and set C is exponential.

Check: Set B's first differences are CAD 3, 5, 7, so they are not constant; its second differences are both

2

. This confirms the quadratic classification.

Common mistakes and how to avoid them

Calling a pattern exponential because its outputs are increasing quickly.

Correction: Check whether consecutive outputs have the same ratio. Fast growth alone does not identify an exponential function.

Calling a table quadratic as soon as the first differences change.

Correction: For equally spaced inputs, check whether the second differences are constant.

Using differences when the input steps are not equal.

Correction: First check the input spacing. The simple difference tests in this lesson assume equal input steps.

Deciding only by a graph's appearance.

Correction: A small portion of a curve can look straight. Check the equation or table pattern when possible.

Lesson summary

Linear functions have constant first differences for equal input steps and graph as straight lines.
Quadratic functions have a squared input in their basic equation and constant second differences for equal input steps.
Exponential functions place the input in the exponent and have a constant ratio between consecutive outputs for equal input steps.
Use equations, tables, and graphs together when possible.

Check your understanding

Question 1

For equally spaced inputs, the outputs are CAD 5, 9, 13, 17. Which type best fits this pattern?

Linear
Quadratic
Exponential
correctIndex,

Show answer and explanation

Linear

Each consecutive difference is

4

. A constant first difference is the linear pattern.

Question 2

A rule is

y=4(3)^x

. Which feature identifies its type?

The input is squared.
The input is in the exponent.
The output has a constant additive increase.
correctIndex,

Show answer and explanation

The input is in the exponent.

The variable

x

appears as an exponent, so the rule is exponential.

Question 3

For equally spaced inputs, a table has first differences CAD 2, 6, 10. Which type best fits the pattern?

Linear
Quadratic
Exponential
correctIndex,

Show answer and explanation

Quadratic

The first differences are not constant, but they increase by

4

each time. The constant second difference indicates a quadratic pattern.

Key terms

Function: A rule that assigns one output to each input.
Difference: The amount found by subtracting one value from another.
Ratio: A comparison found by dividing one value by another.
First difference: The difference between consecutive outputs in a table.
Second difference: The difference between consecutive first differences.
Parabola: The curved graph of a quadratic function.

Continue through MCF3M

View the complete Ontario Grade 11 Mathematics learning path

About this lesson

Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B1.6. It is a study resource, not an official curriculum publication.

Official curriculum reference

Report a correction or ask a question

B1.6 · Distinguish exponential, linear, and quadratic functions

Ontario Grade 11 Mathematics

Use equations, tables, and graph shapes to recognize three important patterns

What you will learn

1. Prerequisite bridge: inputs, outputs, and changes

2. The three patterns in plain language

3. Multiple representations: read the table, then connect it to symbols

4. Guided example and application

5. A reliable decision process

Patterns for equal steps in the input

Classify three patterns from tables

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through MCF3M

About this lesson