Lesson 6 of 7Rates of Change & Combining Functions

Unit 6 · Lesson 6 of 7

Comparing Functions and Choosing a Model

Polynomial, rational, exponential, logarithmic and sinusoidal functions side by side — domain, range, asymptotes, symmetry, end behaviour and growth — and how to choose and justify a model from a description, a table or a graph.

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Key things to remember

1Asymptotes rule out polynomials and sinusoids; vertical asymptotes suggest rational or logarithmic, horizontal ones rational or exponential.
2Exponential: domain all reals, range $y > 0$ (for $y = b^{x}$ ), horizontal asymptote. Logarithmic: domain $x > 0$ , range all reals, vertical asymptote.
3Sinusoidal functions are periodic with a bounded range; polynomials have no asymptotes and end behaviour set by the leading term.
4Any increasing exponential eventually exceeds any polynomial: $2^{x} > x^{3}$ for all $x > 9.94$ (approximately).
5Constant nth differences signal a degree-n polynomial; a constant ratio signals an exponential.
6Justify a model by naming the evidence for it and a feature that rules out the nearest alternative, then check its domain and long-run behaviour in context.

By now you have met five big families of functions, each studied in its own unit. A real problem never tells you which unit it belongs to. A biologist with population counts, an engineer with vibration readings, an economist with costs per item — each starts with data or a description and has to decide which kind of function could describe it. That decision is the most important step in modelling, because no amount of careful algebra rescues the wrong family. Fortunately each family has a fingerprint: a characteristic combination of domain, range, asymptotes, symmetry, end behaviour and pattern of change. This lesson puts those fingerprints side by side, compares how fast the families grow, and turns the comparison into a method for choosing and defending a model.

The five families side by side

Family and example	Domain	Range	Asymptotes	Symmetry	End behaviour
Polynomial, $y = x^{3} - 4 x$	All real x	All real y (odd degree); bounded on one side (even degree)	None	Possible: this one is odd	Like its leading term: here falls to the left, rises to the right
Rational, $y = \frac{1}{x - 2}$	All x except 2	All y except 0	Vertical x = 2; horizontal y = 0	About the point (2, 0)	Approaches the horizontal asymptote
Exponential, $y = 2^{x}$	All real x	y > 0	Horizontal y = 0	None	→ 0 to the left; grows without bound to the right, faster than any polynomial
Logarithmic, $y = lo g_{2} x$	x > 0	All real y	Vertical x = 0	None	Grows without bound, but more slowly than any positive power of x
Sinusoidal, $y = sin x$	All real x	$- 1 \leq y \leq 1$	None	This one is odd; cosine is even	No end behaviour: oscillates forever, period 2π

Transformations move these features (an asymptote shifts with c or d, a range stretches with a), but the fingerprint of each family survives.

Fig 1Parent functions side by side: each family's basic shape is recognizable at a glance.

Fig 2Polynomials have no asymptotes and at most n − 1 turning points, which distinguishes them from every other family.

Read the table as a set of tests. An asymptote rules out polynomials and sinusoids immediately. A vertical asymptote points to rational or logarithmic; a horizontal one to rational or exponential. A domain that stops at a boundary, such as $x > 0$ , suggests a logarithm. A bounded range that repeats suggests a sinusoid. Several turning points with no asymptotes suggest a polynomial, and the number of turns gives a lower bound on its degree. Symmetry is a tiebreaker: polynomials with only odd or only even powers are odd or even, sine is odd, cosine is even, while exponentials and logarithms are never symmetric about the axes.

Growth rates: the exponential always wins in the end

Over a small window, a polynomial can look bigger than an exponential. Compare $x^{3}$ with $2^{x}$ : at $x = 5$ the cubic gives $125$ and the exponential only $32$ . But a polynomial's growth comes from multiplying $x$ by itself a fixed number of times, while an exponential doubles with every step of 1. Once doubling takes over it never lets go: $2^{x}$ passes $x^{3}$ near $x \approx 9.94$ and stays ahead forever. Even $x^{10}$ is caught, near $x \approx 58.8$ . At the other extreme, logarithms grow more slowly than any positive power: $lo g_{2} x$ is only about $4.3$ at $x = 20$ . Two tests make the comparison visible in data. Polynomials have constant $n$ th finite differences; exponentials have a constant ratio between successive values, which is why their differences never settle down.

x	$lo g_{2} x$	$x^{2}$	$x^{3}$	$2^{x}$
2	1	4	8	4
5	2.32	25	125	32
10	3.32	100	1000	1024
20	4.32	400	8000	1 048 576
30	4.91	900	27 000	1 073 741 824

Early on the cubic leads; by x = 10 the exponential has caught it, and by x = 30 the gap is enormous.

Fig 3Constant differences signal a linear or polynomial pattern; a constant ratio signals an exponential one.

VideoCompares a quadratic model with an exponential one and asks which grows faster in the long run.Watch on YouTube

Choosing and justifying a model

"Doubles every", "grows by 3 percent per year", "half-life": a constant ratio per unit time, so exponential.
"Repeats every", "seasonal", "rotates", "high and low tide": a fixed period, so sinusoidal.
"Each tenfold increase adds a fixed amount", decibels, pH, the Richter scale: logarithmic.
"Inversely proportional", "average cost per item", "time to finish shared among workers": a variable in the denominator, so rational, with asymptotes.
Volumes, areas, projectile heights, a rise-fall-rise pattern with no asymptote: polynomial, with degree suggested by the dimensions or the number of turns.

Fig 4The modelling cycle: choose a family from the clues, fit it, check it against the data, and go back if the check fails.

From a table, compute successive differences and successive ratios. Constant second differences point to a quadratic; a constant ratio points to an exponential. From a graph, look for asymptotes, periodicity and turning points. Then justify your choice in a sentence that names the evidence: "The ratios $60 \div 50$ , $72 \div 60$ and $86.4 \div 72$ all equal $1.2$ , so an exponential model $y = 50 (1.2)^{t}$ fits." Finally, test the model's domain and long-run behaviour against the situation. A population model that predicts negative animals, or a cost model that never levels off when costs obviously must, is the wrong family even if it fits the data points you were given.

Fig 5The logarithm's fingerprint: a vertical asymptote at x = 0, an x-intercept at 1, and slow, unbounded growth.

A model is a claim that needs evidence

Say which family you chose, which features of the data or description support it, and one feature that rules out the closest alternative. For example: "Exponential, because the ratios are constant; not quadratic, because the second differences keep growing." A choice with no evidence is a guess.

VideoAn Ontario Grade 12 lesson on modelling real situations with functions.Watch on YouTube

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Compare

f (x) = x^{3} - 4 x

g (x) = \frac{1}{x - 2}

and

h (x) = 3 (0.5)^{x}

: domain, range, asymptotes, symmetry and end behaviour.

Example 2

Between which two consecutive integers does

2^{x}

overtake

x^{3}

for good? Confirm with technology.

Example 3

Choose a model for each table. Table A at

t = 0, 1, 2, 3, 4

50, 60, 72, 86.4, 103.68

. Table B at

x = 1, 2, 3, 4, 5

4, 13, 28, 49, 76

Example 4

Choose and justify a family for each: (a) a medication's concentration halves every 6 hours; (b) average cost per item when fixed costs are 5000 dollars and each item costs 2 dollars to make; (c) a sound's loudness in decibels as its intensity increases; (d) hourly outdoor temperature over a week.

Where marks get lost

Choosing a quadratic for growth data because the first few points curve upward, without checking whether the ratios are constant.
Assuming a polynomial with a large coefficient, such as $1000 x^{2}$ , will always beat an exponential. Any increasing exponential eventually wins.
Confusing the domains of exponentials and logarithms: $2^{x}$ accepts every $x$ ; $lo g_{2} x$ needs $x > 0$ .
Calling every curve with an asymptote exponential, when rational and logarithmic functions have asymptotes too.
Justifying a model only by saying it looks right, with no differences, ratios or features cited as evidence.

Flashcards 9

Families with no asymptotes

Family with a vertical asymptote and domain x > 0

Family with both vertical and horizontal asymptotes

Range of

y = b^{x}

+ 5 more

Practice

A function has domain all real numbers, range y > 0 and a horizontal asymptote. Which family is it most likely from?

A graph has a vertical asymptote at x = 3 and a horizontal asymptote at y = 1. Which family fits?

Find the second differences of the sequence

4, 13, 28, 49, 76

Find the common ratio of the data

50, 60, 72, 86.4

Beyond what value of

x

2^{x} > x^{10}

for good? Use technology and give 1 decimal place.

Order these values at

x = 20

from smallest to largest.

1 $x^{2}$
2 $2^{x}$
3 $x^{3}$
4 $lo g_{2} x$

Each whole-number increase on the Richter scale represents a tenfold increase in wave amplitude. Which function family models magnitude in terms of amplitude?

Match each situation to its most suitable model family.

Average cost per item as production grows

Height of a tide over two days

A bank balance growing at 4 percent per year

Volume of an open box as the corner cut changes

Which function is odd?

A student fits

y = 0.8 x^{2} + 2 x + 50

to population data that turn out to grow by a constant 20 percent per year. What is the strongest objection?

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