D3.1 Compare major families of functions | Ontario Grade 12 Mathematics

Ontario Grade 12 Mathematics

Characteristics of Functions

Use equations, graphs, tables, and numerical rates of change to compare functions

Functions can rise, fall, turn, repeat, or approach a line. These patterns help us compare them. A repeating wave suggests a trigonometric function. A parabola suggests a quadratic polynomial. A graph alone may not identify every detail, so we also use equations and values. Comparing function families means describing similarities and differences with evidence. This lesson uses equations, graphs, tables, and numerical rates of change.

What you will learn

Recognize polynomial, rational, exponential, logarithmic, and trigonometric functions from their equations.
Compare function families using domains, ranges, intercepts, graph shapes, and patterns of change.
Calculate and interpret average rates of change over a stated interval.
Support comparisons with evidence from equations, graphs, or tables.

1. Prerequisite bridge: inputs, outputs, and rates

A function assigns one output to each allowed input. The input is often written as

x

, and the output as

f(x)

. The domain is the set of allowed inputs. The range is the set of outputs the function can produce. For example,

f(x)=x^2

has all real numbers as its domain, and its range is

y\geq 0

.

An intercept is where a graph meets an axis. To find a

y

-intercept, use an input of zero. An

x

-intercept occurs where the output is zero. These features help compare graphs, but two functions can share an intercept and still have different shapes.

A rate of change describes how much an output changes as the input changes. The average rate of change between two inputs is the output change divided by the input change. On a graph, it is the slope of the line joining the two points. The interval matters: a curve can have different average rates on different intervals.

Before drawing a conclusion from a graph, note the interval shown. A curve may look almost straight over a short interval but bend more over a wider one. A repeating function may not show a full cycle in a small viewing window.

\frac{f(b)-f(a)}{b-a}

Use domain and range to describe possible inputs and outputs.
Use intercepts and graph shape as evidence, not as the only evidence.
State the interval when comparing average rates of change.

2. Recognize the major families

A polynomial function combines powers of the input with non-negative whole-number exponents. Linear and quadratic functions are examples. A linear graph is a straight line with a constant rate of change. A quadratic graph is a parabola with a turning point. Polynomials with higher powers can have more turns, depending on their terms and coefficients.

A rational function is a quotient of polynomials. Its denominator cannot be zero, so some inputs may be excluded. Its graph may have an asymptote: a line that the graph approaches in a particular part of its behaviour. Check the equation and graph before making claims about asymptotes.

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

f(x)=a^x

, where

a>0

and

a\neq 1

. Its outputs are positive. If

a>1

, it increases; if

0<a<1

, it decreases. For equal input steps, its outputs change by a constant factor rather than a constant amount.

A logarithmic function reverses an exponential relationship. In the basic function

f(x)=\log_a x

, inputs must be positive, with

a>0

and

a\neq 1

. Its graph passes through

(1,0)

and has a vertical asymptote at

x=0

. If

a>1

, the function increases, with less rapid growth as

x

increases.

A trigonometric function such as sine or cosine repeats its values. A function with a repeating pattern is periodic; the period is the input length of one complete cycle. The basic sine and cosine functions have outputs from

-1

to

1

. Changes to their equations can shift or stretch their graphs, so check the specific function.

Look at the role of the input in the equation to identify a family.
Rational and logarithmic functions can have restrictions on their inputs.
Exponential functions change by a factor over equal input steps; linear functions change by a constant amount.
Basic sine and cosine functions repeat, while polynomial functions do not repeat in that way.

3. Compare equations, tables, and graphs

A useful comparison combines three representations. An equation can identify a family and reveal input restrictions. A table shows how outputs respond to selected inputs. A graph shows overall shape, intercepts, turning points, repeating behaviour, and asymptotes.

Consider

p(x)=x^2

and

q(x)=2^x

. In the first equation, the input is squared. In the second, the input is the exponent. Both functions are defined for every real input, but their values differ. At

x=1

, the outputs are

1

and

2

. At

x=3

, they are

9

and

8

. A shared or nearby value at one input does not mean the functions follow the same rule.

The average rates of change from

x=1

to

x=3

can be compared fairly because the interval is the same. For the quadratic, the output changes from

1

to

9

, a change of

8

. For the exponential, it changes from

2

to

8

, a change of

6

. Divide each output change by the input change of

2

. The quadratic’s average rate is

4

, and the exponential’s is

3

. This conclusion applies to this interval only.

A strong comparison includes a similarity and a difference. Both functions increase over the chosen interval. Their average rates differ, and their graph patterns differ: the quadratic is symmetric about the vertical axis, while the exponential is not. The equation, values, and graph each provide useful evidence.

\frac{p(3)-p(1)}{3-1}=4,\quad \frac{q(3)-q(1)}{3-1}=3

Compare rates over the same interval.
A conclusion about one interval does not describe every input.
Use more than one representation when the comparison needs stronger evidence.

4. Apply comparisons carefully

For two unfamiliar functions, first read their equations. Identify the families and any restrictions on inputs. Then inspect a graph or make a small table. Compare features that apply, such as intercepts, range, symmetry, repeating behaviour, and average rate of change.

Do not assume every function in a family has the exact features of its basic example. A shifted quadratic may have a turning point away from the origin. A changed sine function may have a different range. A rational function may have different asymptotes depending on its equation. Use the specific equation or graph as evidence.

Make each comparison precise. Name the functions, state a similarity, and explain at least one difference. If you compare rates, give the interval. If you refer to a graph, identify visible evidence such as a turning point, repeating cycle, or asymptote.

Check the specific equation before describing a graph feature.
Support comparisons with a table value, equation feature, graph feature, or interval-based rate.

Clues for comparing function families

Family	Clue in the equation	Graph feature to check
Polynomial	Non-negative whole-number powers of the input	May be a line, a parabola, or a curve with turns
Rational	A quotient of polynomials	Check excluded inputs and possible asymptotes
Exponential	The input is the exponent	Changes by a constant factor over equal input steps
Logarithmic	A logarithm of the input	Basic form requires positive inputs and has an asymptote at the vertical axis
Trigonometric	A sine, cosine, or other trigonometric rule	Check for a repeating pattern

Worked example

Compare a quadratic and an exponential function

Compare

p(x)=x^2

and

q(x)=2^x

from

x=1

to

x=3

. Describe a similarity, compare their average rates of change, and identify a difference in their graph patterns.

Identify the families
In $p(x)=x^2$ , the input is raised to a whole-number power, so $p$ is a polynomial function. In $q(x)=2^x$ , the input is the exponent, so $q$ is an exponential function.
Find endpoint values
Substitute each endpoint into both rules. At $x=1$ , the quadratic gives $1$ , while the exponential gives $2$ . At $x=3$ , the outputs are $9$ and $8$ . Both functions increase from the first endpoint to the second.
$p(1)=1,\quad p(3)=9,\quad q(1)=2,\quad q(3)=8$
Compare average rates
The input change is $3-1=2$ . Divide each function’s output change by this same amount. The quadratic’s average rate is $4$ , and the exponential’s is $3$ .
$\frac{p(3)-p(1)}{3-1}=4,\quad \frac{q(3)-q(1)}{3-1}=3$
Compare graph patterns
The quadratic graph is symmetric about the vertical axis and has a turning point at $(0,0)$ . The exponential graph is increasing and is not symmetric about that axis. The rate comparison applies only to the stated interval.

Answer: Both functions increase from

x=1

to

x=3

. The quadratic’s average rate of change is

4

, compared with

3

for the exponential on this interval. Their graph patterns differ: the quadratic is symmetric about the vertical axis, while the exponential is not.

Check: For the quadratic, the output change is

9-1=8

, so its average rate is

8/2=4

. For the exponential, the output change is

8-2=6

, so its average rate is

6/2=3

.

Common mistakes and how to avoid them

Claiming that the function with the larger average rate on one interval always grows faster.

Correction: State the interval and limit the claim to it. Compare another interval before making a broader statement.

Assuming every function in a family has the exact shape of its basic example.

Correction: Check the specific equation or graph. Changes to an equation can alter important features.

Treating an asymptote as a line the graph can never touch.

Correction: An asymptote describes a line the graph approaches in a particular part of its behaviour. Check the specific function before claiming whether it meets that line.

Lesson summary

Compare equations, graphs, and values to identify families and their features.
Use domain, range, intercepts, symmetry, periodic behaviour, and asymptotes when they apply.
Use average rates of change over the same stated interval for numerical comparisons.
Support conclusions with evidence and do not extend an interval-based result too far.

Check your understanding

Question 1

Which function has the input in the exponent?

$f(x)=x^3$
$g(x)=\frac{1}{x+2}$
$h(x)=3^x$
$k(x)=\log_3 x$

Show answer and explanation

h(x)=3^x

In

h(x)=3^x

, the input

x

is the exponent, so

h

is exponential.

Question 2

For

f(x)=x^2

and

g(x)=2^x

, which statement about the interval from

x=1

to

x=3

is correct?

Both average rates are $2$ .
The average rate for $f$ is $4$ and for $g$ is $3$ .
The average rate for $f$ is $3$ and for $g$ is $4$ .
The functions have the same outputs at both endpoints.

Show answer and explanation

The average rate for

f

is

4

and for

g

is

3

.

The endpoint outputs are

1

and

9

for

f

, and

2

and

8

for

g

. Dividing each output change by the input change of

2

gives

4

and

3

.

Key terms

Domain: The set of inputs allowed for a function.
Range: The set of outputs a function can produce.
Average rate of change: The output change divided by the input change between two points.
Asymptote: A line that a graph approaches in a particular part of its behaviour.
Periodic: Repeating the same pattern after a fixed input interval.

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Published by DoAssignment. This AI-assisted lesson follows Ontario Grade 12 Mathematics (MHF4U), expectation D3.1. It is a study resource, not an official curriculum publication.

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D3.1 · Compare major families of functions