Distinguish exponential, linear, and quadratic functions

Ontario Grade 11 Mathematics

Exponential Functions

Recognising Each Function Family by Its Equation, Table, and Graph

Three function families appear constantly in mathematics and real life: linear, quadratic, and exponential. A linear function might model a taxi fare that grows by a fixed amount per kilometre. A quadratic might describe the height of a ball thrown into the air. An exponential might track a savings account that grows by a fixed percentage each year. All three involve an input and an output, yet they behave in fundamentally different ways. Learning to tell them apart — from an equation, a table, or a graph — is one of the most important skills in this course. This lesson builds that skill step by step, starting with a quick review of what you already know about each family.

What you will learn

Identify whether a function is linear, quadratic, or exponential from its equation.
Use a table of values to distinguish the three function families by examining how outputs change.
Describe the key graphical features that separate each family.
Apply these distinctions to realistic contexts and mixed examples.

Prerequisite Bridge: What Makes a Function?

A function is a rule that assigns exactly one output value to each input value. We often write this using function notation:

f(x)

means 'the output of function

f

when the input is

x

'. For example, if

f(x) = 2x + 1

, then

f(3) = 2(3) + 1 = 7

.

In Grade 10 you met linear functions (straight-line graphs) and quadratic functions (parabola-shaped graphs). In this course you will also work extensively with exponential functions. Before sorting them by their differences, it helps to recall their basic forms: a linear function has the form

f(x) = mx + b

, a quadratic has the form

f(x) = ax^2 + bx + c

where

a \neq 0

, and an exponential has the form

f(x) = a \cdot b^x

where

b > 0

and

b \neq 1

.

A function gives exactly one output for each input.
Function notation $f(x)$ names both the function and the input in one expression.
Linear: $f(x) = mx + b$ ; Quadratic: $f(x) = ax^2 + bx + c$ ; Exponential: $f(x) = a \cdot b^x$ .

Identifying Each Family from Its Equation

The equation is usually the fastest way to classify a function. Ask yourself: where does the variable

x

appear?

In a linear function,

x

appears only to the first power and is never in an exponent. Examples:

f(x) = 5x - 3

and

g(x) = -2x + 7

. The highest power of

x

is

1

.

In a quadratic function, the highest power of

x

is

2

, and there is no

x

in an exponent position. Examples:

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

and

g(x) = -x^2 + 9

. The squared term is what gives the parabola its shape.

In an exponential function,

x

is the exponent on a constant base. Examples:

f(x) = 2^x

and

g(x) = 3 \cdot (1.5)^x

. Notice that the base is a fixed positive number, not

1

, and the variable is 'up in the power'. This is completely different from

x^2

, where the base changes and the power is fixed.

Linear: $x$ is to the power of $1$ only.
Quadratic: highest power of $x$ is $2$ ; $x$ is never in the exponent.
Exponential: $x$ is the exponent; the base is a fixed positive constant.
A function like $f(x) = x^2$ has a fixed exponent (2) and a variable base — that is quadratic, not exponential.

Identifying Each Family from a Table of Values — Differences and Ratios

When you have a table of evenly spaced input values, you can classify the function by looking at how the outputs change. This technique uses what are called first differences, second differences, and ratios.

First differences are found by subtracting each output from the next one:

\Delta y = y_{n+1} - y_n

. If the first differences are all equal (constant), the function is linear. The constant first difference equals the slope

m

.

If the first differences are not constant, calculate the second differences: subtract each first difference from the next. If the second differences are all equal and non-zero, the function is quadratic. Constant second differences are the hallmark of a quadratic.

If neither first nor second differences are constant, check the ratio of consecutive outputs:

r = \frac{y_{n+1}}{y_n}

. If this ratio is the same for every consecutive pair, the function is exponential. That constant ratio equals the base

b

in

f(x) = a \cdot b^x

.

This test only works reliably when the input values are equally spaced (e.g.,

x = 0, 1, 2, 3, \ldots

). Always check that spacing first.

r = \frac{y_{n+1}}{y_n}

Constant first differences → linear function.
Constant second differences (non-zero) → quadratic function.
Constant ratio of consecutive outputs → exponential function.
Input values must be equally spaced for this test to be valid.

Identifying Each Family from Its Graph

Each function family has a distinctive graph shape. A linear function produces a straight line. Its direction depends on the slope: positive slope goes up left-to-right, negative slope goes down. The graph extends infinitely in both directions without bending.

A quadratic function produces a parabola — a smooth, symmetric U-shape (or upside-down U if

a < 0

). The parabola has a single turning point called the vertex. One side mirrors the other across the axis of symmetry.

An exponential function produces a curve that either rises steeply without bound or decays toward the x-axis without ever touching it. When

b > 1

the graph grows (exponential growth); when

0 < b < 1

the graph decays (exponential decay). The x-axis is a horizontal asymptote — the curve approaches it but never crosses it.

A quick visual check: if it bends once and is symmetric, it is quadratic. If it bends but is not symmetric and heads steeply up or flattens toward an axis, it is exponential. If it does not bend at all, it is linear.

Linear graph: straight line, no bending.
Quadratic graph: symmetric parabola with one vertex.
Exponential graph: curve that grows steeply or decays toward a horizontal asymptote.
Asymptote: a line the curve approaches but never reaches.

Putting It Together — Comparing Long-Run Behaviour

One of the most useful practical distinctions is how fast each family grows as

x

increases. For large positive values of

x

, exponential growth eventually overtakes both linear and quadratic growth, even if it starts out smaller.

Consider the functions

f(x) = 100x

(linear),

g(x) = x^2

(quadratic), and

h(x) = 2^x

(exponential). At

x = 10

:

f(10) = 1000

,

g(10) = 100

,

h(10) = 1024

. At

x = 20

:

f(20) = 2000

,

g(20) = 400

,

h(20) = 1048576

. The exponential has pulled far ahead.

This long-run behaviour matters in real contexts: a population growing at a fixed percentage each year will eventually exceed any linearly or quadratically growing quantity, even if the linear model looks larger at first. Recognising which model applies helps you make better predictions.

For large $x$ , exponential growth surpasses both linear and quadratic growth.
Linear functions grow by equal amounts per step; quadratic functions grow by increasing amounts; exponential functions grow by increasing percentages.
Choosing the right function family leads to more reliable predictions.

Summary: Three Ways to Distinguish the Function Families

Feature	Linear	Quadratic	Exponential
General equation	$f(x) = mx + b$	$f(x) = ax^2 + bx + c,\; a \neq 0$	$f(x) = a \cdot b^x,\; b > 0,\; b \neq 1$
Where is $x$ ?	Base, power 1	Base, power 2	In the exponent
Table test	Constant 1st differences	Constant 2nd differences	Constant ratio of outputs
Graph shape	Straight line	Symmetric parabola	Steep curve with horizontal asymptote
Long-run growth	Grows by equal steps	Grows by increasing steps	Eventually overtakes both others

Worked example

Classifying Four Functions from Their Equations and a Table

Classify each of the following as linear, quadratic, or exponential. Justify your answer.
(a)

f(x) = 4x - 9

(b)

g(x) = 5 \cdot 3^x

(c)

h(x) = -2x^2 + 6x

(d) The table below, where inputs are equally spaced:

x

: 0, 1, 2, 3, 4

y

: 3, 6, 12, 24, 48

Classify part (a)
Look at where $x$ appears in $f(x) = 4x - 9$ . The variable $x$ is raised only to the power $1$ , and there is no $x$ in an exponent. This matches the linear form $mx + b$ with $m = 4$ and $b = -9$ .
$f(x) = 4x - 9$
Classify part (b)
In $g(x) = 5 \cdot 3^x$ , the variable $x$ is in the exponent position on the base $3$ . The base is a fixed positive constant not equal to $1$ . This matches the exponential form $a \cdot b^x$ with $a = 5$ and $b = 3$ .
$g(x) = 5 · 3^x$
Classify part (c)
In $h(x) = -2x^2 + 6x$ , the highest power of $x$ is $2$ , and $x$ does not appear as an exponent on a fixed base. The coefficient of $x^2$ is $-2 \neq 0$ . This matches the quadratic form $ax^2 + bx + c$ with $a = -2$ , $b = 6$ , and $c = 0$ .
$h(x) = -2x^2 + 6x$
Compute first differences for the table
Subtract each $y$ -value from the one that follows it. The first differences are $6-3=3$ , $12-6=6$ , $24-12=12$ , $48-24=24$ . These are not equal, so the function is not linear.
$\Delta y: 3, 6, 12, 24$
Check second differences
Subtract each first difference from the next: $6-3=3$ , $12-6=6$ , $24-12=12$ . The second differences are CAD 3, 6, 12 — not equal. So the function is not quadratic.
$\Delta^2 y: 3, 6, 12$
Check the ratio of consecutive outputs
Divide each $y$ -value by the one before it: $6 \div 3 = 2$ , $12 \div 6 = 2$ , $24 \div 12 = 2$ , $48 \div 24 = 2$ . The ratio is constant at $2$ every time. This confirms the table represents an exponential function with base $b = 2$ .
$r = \frac{6}{3} = \frac{12}{6} = \frac{24}{12} = \frac{48}{24} = 2$

Answer: (a) Linear. (b) Exponential. (c) Quadratic. (d) Exponential with base

b = 2

.

Check: For part (d), if the function is

y = 3 \cdot 2^x

, then at

x = 0

:

3 \cdot 2^0 = 3

✓; at

x = 3

:

3 \cdot 2^3 = 3 \cdot 8 = 24

✓. All table values are confirmed.

Worked example

Sorting Three Tables and Choosing the Right Model

Three tables are given, each with equally spaced inputs from

x = 0

to

x = 4

. Determine which table represents a linear function, which represents a quadratic function, and which represents an exponential function. Explain your reasoning using differences or ratios.

Table A —

y

values: 2, 5, 8, 11, 14
Table B —

y

values: 1, 3, 9, 27, 81
Table C —

y

values: 0, 1, 4, 9, 16

Compute first differences for Table A
Subtract each output from the next: $5-2=3$ , $8-5=3$ , $11-8=3$ , $14-11=3$ . All first differences equal $3$ . Constant first differences mean the function is linear.
$\Delta y_A: 3, 3, 3, 3$
Identify Table A
Because the first differences are constant and equal to $3$ , Table A is linear with slope $m = 3$ . The starting value at $x = 0$ is $y = 2$ , so the equation is $f(x) = 3x + 2$ .
$f(x) = 3x + 2$
Compute first differences for Table B
First differences: $3-1=2$ , $9-3=6$ , $27-9=18$ , $81-27=54$ . These are not constant, so Table B is not linear.
$\Delta y_B: 2, 6, 18, 54$
Check ratios for Table B
Divide each output by the previous one: $3 \div 1 = 3$ , $9 \div 3 = 3$ , $27 \div 9 = 3$ , $81 \div 27 = 3$ . The ratio is constantly $3$ , confirming Table B is exponential with base $b = 3$ and initial value $a = 1$ .
$r = \frac{3}{1} = \frac{9}{3} = \frac{27}{9} = \frac{81}{27} = 3$
Compute first and second differences for Table C
First differences: $1-0=1$ , $4-1=3$ , $9-4=5$ , $16-9=7$ . These are not constant. Second differences: $3-1=2$ , $5-3=2$ , $7-5=2$ . All second differences equal $2$ , which is non-zero and constant. Table C is quadratic.
$\Delta y_C: 1, 3, 5, 7 \Delta^2 y_C: 2, 2, 2$
Identify Table C's equation
The outputs CAD 0, 1, 4, 9, 16 are the perfect squares $0^2, 1^2, 2^2, 3^2, 4^2$ , which confirms the equation is $f(x) = x^2$ . The constant second difference of $2$ equals $2a$ , giving $a = 1$ , which matches.
$f(x) = x^2$

Answer: Table A is linear:

f(x) = 3x + 2

. Table B is exponential:

f(x) = 3^x

. Table C is quadratic:

f(x) = x^2

.

Check: Table B at

x = 4

:

3^4 = 81

✓. Table C at

x = 4

:

4^2 = 16

✓. Table A at

x = 4

:

3(4) + 2 = 14

✓.

Common mistakes and how to avoid them

Confusing

x^2

(quadratic) with

2^x

(exponential) because both involve the numbers

x

and

2

.

Correction: Ask: is

x

the base or the exponent? In

x^2

the variable is the base and

2

is the fixed power — quadratic. In

2^x

the variable is the exponent and

2

is the fixed base — exponential.

Applying the differences test when the input values are not equally spaced, leading to a wrong classification.

Correction: Always check that consecutive

x

-values are evenly spaced (e.g., increase by

1

each time) before computing first differences, second differences, or ratios.

Concluding a function is quadratic after seeing that first differences are not constant, without checking second differences.

Correction: Non-constant first differences rule out linear, but you must check second differences for quadratic and then check ratios for exponential before deciding.

Thinking

f(x) = 1^x

is exponential because

x

is in the exponent.

Correction: When the base equals

1

, the output is always

1

regardless of

x

, so the function is actually constant. The definition of exponential requires

b \neq 1

.

Misreading the graph of an exponential decay as linear because the curve looks nearly straight for small values of

x

.

Correction: Check the table or equation. A true linear function produces exactly equal first differences; an exponential always produces a constant ratio, which you can verify numerically.

Lesson summary

A linear function has $x$ to the first power and produces a straight-line graph; its table shows constant first differences.
A quadratic function has $x$ to the second power as the highest term and produces a symmetric parabola; its table shows constant second differences.
An exponential function has $x$ in the exponent on a fixed positive base and produces a steeply rising or decaying curve; its table shows a constant ratio between consecutive outputs.
When reading an equation, focus on where $x$ appears — the base or the exponent — to classify the function instantly.
For equally spaced inputs, the differences-and-ratios test is a reliable tool: constant 1st differences → linear; constant 2nd differences → quadratic; constant ratio → exponential.
For large values of $x$ , exponential growth will always eventually outpace both linear and quadratic growth.

Check your understanding

Question 1

Which of the following equations represents an exponential function?

$f(x) = 7x^2 - 3$
$f(x) = 4x + 1$
$f(x) = 6 \cdot 2^x$
$f(x) = x^3 + 2x$

Show answer and explanation

f(x) = 6 \cdot 2^x

f(x) = 6 \cdot 2^x

has

x

in the exponent on the fixed base

2

, which is the defining feature of an exponential function. Option A is quadratic, option B is linear, and option D has a highest power of

3

(cubic — not one of our three families in this lesson).

Question 2

A table of values with equally spaced inputs gives first differences of CAD 5, 5, 5, 5. What type of function does this table most likely represent?

Quadratic, because the differences are positive
Exponential, because the differences are equal
Linear, because the first differences are constant
Quadratic, because you need to check second differences next

Show answer and explanation

Linear, because the first differences are constant

Constant first differences are the signature of a linear function. There is no need to check further — if every consecutive pair of outputs differs by the same amount, the relationship is linear. The slope equals that constant difference.

Question 3

A table with equally spaced inputs gives the output values 4, 8, 16, 32, 64. Which test confirms this is exponential?

The first differences are all equal to 4
The second differences are all equal to 8
Each output divided by the previous output gives a constant ratio of 2
The outputs increase, which only happens for exponential functions

Show answer and explanation

Each output divided by the previous output gives a constant ratio of 2

Dividing consecutive outputs:

8 \div 4 = 2

,

16 \div 8 = 2

,

32 \div 16 = 2

,

64 \div 32 = 2

. A constant ratio of

2

confirms exponential behaviour with base

b = 2

. Note that the first differences (4, 8, 16, 32) are not constant, ruling out linear, and the second differences are also not constant, ruling out quadratic.

Question 4

The graph of a function is a smooth U-shaped curve that is symmetric about a vertical line. Which function family does this graph belong to?

Linear, because symmetry means the slope is constant
Exponential, because the curve bends upward
Quadratic, because the graph is a symmetric parabola with one turning point
Exponential, because it is not a straight line

Show answer and explanation

Quadratic, because the graph is a symmetric parabola with one turning point

A symmetric U-shaped curve with a single turning point (vertex) is a parabola, which is the graph of a quadratic function. Exponential graphs also curve upward, but they are not symmetric — one side flattens toward an asymptote while the other rises steeply.

Key terms

Linear function: A function of the form $f(x) = mx + b$ , where $x$ appears only to the first power. Its graph is a straight line.
Quadratic function: A function of the form $f(x) = ax^2 + bx + c$ with $a \neq 0$ . Its graph is a symmetric parabola.
Exponential function: A function of the form $f(x) = a \cdot b^x$ where $b > 0$ and $b \neq 1$ . The variable $x$ appears as the exponent on a fixed base.
First difference: The result of subtracting a table output from the next one: $\Delta y = y_{n+1} - y_n$ . Constant first differences indicate a linear function.
Second difference: The result of subtracting a first difference from the next first difference. Constant (non-zero) second differences indicate a quadratic function.
Common ratio: The constant value obtained by dividing any output by the previous output in an exponential table. It equals the base $b$ of the exponential function.
Asymptote: A line that a curve approaches but never reaches. Exponential functions have a horizontal asymptote along the x-axis (or a line parallel to it).
Vertex: The single turning point of a parabola — the highest or lowest point of the quadratic graph, lying on the axis of symmetry.

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