C. Polynomial and Rational Functions · Polynomial Functions and Equations

Polynomial, Exponential or Sinusoidal? Telling Function Families Apart

C1.4 C1.2About 60 minutes

Learning goals

I can tell polynomial, exponential and sinusoidal functions apart from their equations.
I can classify a table of values using finite differences, ratios or repeating values.
I can tell the families apart from their graphs using end behaviour, turning points, asymptotes and range.
I can explain why an exponential function eventually grows faster than any polynomial.

Words to know

exponential function
A function such as y = a(b)ˣ, b > 0, b ≠ 1, where the variable is in the exponent.

sinusoidal function
A function with the shape of y = sin x, such as y = a sin(k(x − d)) + c. Its values repeat in cycles.

periodic
Repeating its values at regular intervals of x. The length of one cycle is the period.

common ratio
For equally spaced x-values, the constant quotient y₂/y₁ = y₃/y₂ = ... found in exponential data.

horizontal asymptote
A horizontal line that the graph approaches more and more closely at one end without reaching it.

Three families, three fingerprints

You have now met three big families of functions:

Polynomial (this unit): y = x³ − 2x, y = −x⁴ + 5, ...
Exponential (Grade 11 Exponential Functions, and the MHF4U unit Exponential and Logarithmic Functions): y = 3(2)ˣ, y = 100(0.5)ˣ, ...
Sinusoidal (Grade 11 Sinusoidal Functions, and the MHF4U unit Radian Measure and Trigonometric Functions): y = 2 sin(1.5x), y = 4 cos x + 1, ...

In the equation, look at where the x is. In a polynomial, x is the base and the exponents are fixed whole numbers (x⁴). In an exponential function, x is the exponent (4ˣ). In a sinusoidal function, x is inside sin or cos.

One example from each family. The polynomial has a few turns and runs off to infinity at both ends. The exponential never turns and hugs its asymptote on one side. The sinusoidal function turns again and again and stays between two horizontal lines.

Feature	Polynomial (degree ≥ 1)	Exponential y = a(b)ˣ	Sinusoidal
Table (equal x-steps)	nth differences constant	ratios y₂/y₁ constant	values repeat in cycles
End behaviour	at least one end → ±∞	→ ±∞ at one end, → asymptote at the other	keeps oscillating, no limit
Turning points	finite: at most n − 1	none	infinitely many
Zeros	at most n	none (for y = a(b)ˣ)	infinitely many (if the axis is crossed)
Range	all reals (odd n) or y ≥ min / y ≤ max (even n)	y > 0 if a > 0	between a minimum and a maximum
Asymptote	none	horizontal	none

How to tell the three families apart.

Worked example

Example 1: Classify from tables

Each table uses x = 0, 1, 2, 3, 4, 5. Is the function polynomial, exponential or sinusoidal?

(A) y = 1, 3, 9, 27, 81, 243 (B) y = 2, 5, 2, −1, 2, 5 (C) y = 2, 2, 2, 8, 26, 62

(A) Ratios: 3/1 = 9/3 = 27/9 = ... = 3. A constant ratio means exponential (y = 3ˣ). Its differences 2, 6, 18, 54, 162 are themselves tripling and never settle.
(B) The values go 2, 5, 2, −1 and then start again at 2, 5. Repeating values suggest sinusoidal (period 4, between −1 and 5).
(C) Differences: 0, 0, 6, 18, 36; then 0, 6, 12, 18; then 6, 6, 6. Constant third differences: cubic polynomial (6a = 6, a = 1). The flat start is just a coincidence of this particular cubic, y = x³ − 3x² + 2x + 2.

(A) exponential (B) sinusoidal (C) polynomial (cubic)

Worked example

Example 2: Classify from features

Which family fits each description?

(a) Has a horizontal asymptote y = 4 on the left and increases everywhere. (b) Range −3 ≤ y ≤ 7; turning points every 6 units. (c) Both ends go down; three turning points. (d) Equation y = 2x⁵ − x.

(a) An asymptote and no turning points: exponential (for example y = 2ˣ + 4).
(b) Bounded range and endless regular turning: sinusoidal.
(c) A finite number of turning points and ends going to −∞: polynomial (even degree, at least 4, with a < 0).
(d) x is the base and 5 is a fixed whole-number exponent: polynomial (quintic).

(a) exponential (b) sinusoidal (c) polynomial (d) polynomial

Worked example

Example 3: The race between x³ and 2ˣ

Compare y = x³ and y = 2ˣ for x = 1, 2, 5, 9, 10 and 12. Which grows faster in the long run?

x = 1: 2ˣ = 2, x³ = 1 (exponential ahead).
x = 2: 4 and 8; x = 5: 32 and 125; x = 9: 512 and 729 (cubic ahead).
x = 10: 1024 and 1000; x = 12: 4096 and 1728 (exponential ahead, and pulling away).
Each step of 1 multiplies 2ˣ by 2, but multiplies x³ by ((x + 1)/x)³, which gets closer and closer to 1 as x grows. So once 2ˣ is ahead for large x, it stays ahead.

2ˣ wins in the long run. In fact, any exponential function with b > 1 eventually grows faster than any polynomial.

y = x³ (solid) is above y = 2ˣ (dashed) for about 1.37 < x < 9.94, but the exponential overtakes near x ≈ 9.94 and never looks back.

Just for fun

In 2002, a high-school student named Britney Gallivan folded one very long strip of paper in half 12 times, giving 2¹² = 4096 layers. Each fold doubles the thickness, which is exactly why each fold is so much harder than the one before: exponential growth in action.

Hands-on activity

Three experiments, three families

You need: a sheet of paper, a ruler, an empty soup can (no sharp edges; a grown-up can check), a small piece of tape and a pencil.

Folding: fold the paper in half again and again. After each fold record the number of layers. (Stop when it gets too hard.)
Rolling: put a small piece of tape on the rim of the can, starting at the bottom. Roll the can a quarter turn at a time along a table and measure the height of the tape above the table each time, for at least 2 full turns.
Boxes: use your box volumes from lesson 1, or calculate V(x) = x(27.9 − 2x)(21.6 − 2x) for x = 1, 2, 3, 4, 5.

Make a table for each experiment and decide which family fits by checking ratios, repeating values or differences.

What to look for: folding gives 2, 4, 8, 16, ... (constant ratio 2: exponential). The tape height goes 0, r, 2r, r, 0, r, ... where r is the can's radius (repeats: sinusoidal). The box volumes have constant third differences (polynomial).

Your turn

x = 0, 1, 2, 3, 4 and y = 4, 12, 36, 108, 324. Which family?

x = 0, 1, 2, 3, 4, 5, 6 and y = 0, 3, 0, −3, 0, 3, 0. Which family?

x = 0, 1, 2, 3, 4 and y = 1, 2, 9, 28, 65. Which family?

Which of the three families can have a horizontal asymptote?

Which family can have infinitely many zeros?

Classify y = x⁵, y = 5ˣ and y = 5 sin x.

Explain why no polynomial of degree 1 or more can have range −2 ≤ y ≤ 2.

More practice: warm-up

Which family is periodic?

Constant ratios in a table (equal x-steps) point to which family?

Constant third differences point to which family?

Does y = 2(3)ˣ have any turning points?

Is y = x³ an exponential function?

More practice: on your own

x = 0, 1, 2, 3, 4 and y = 100, 50, 25, 12.5, 6.25. Classify and write the equation.

x = 0, 1, 2, 3, 4, 5 and y = 0, 1, 16, 81, 256, 625. Classify and give the degree.

Compare 1.1ˣ and x² at x = 10, 50 and 100.

A function has range y > 2 and decreases everywhere. Which family? Give one possible equation.

A graph has turning points at x = 1, 3, 5, 7, 9, ... and keeps going. Could it be a polynomial?

Challenge

Between which two consecutive whole numbers does 2ˣ overtake x³ for good? Show the values.

Show that the first differences of y = a(b)ˣ (x-step 1) form another exponential pattern with the same ratio b. Why does this mean they are never constant when b ≠ 1?

Daily high temperatures in Thunder Bay were recorded for three years. Would a polynomial be a good model for predicting next year? Explain.

Check your understanding

Quiz

Which function is exponential?

A table with equal x-steps has values 5, 10, 20, 40, 80. What kind of function fits?

Which feature belongs to sinusoidal functions but not to polynomial functions?

Which feature does y = 3(2)ˣ have?

For large positive x, which is largest?

A graph has 2 turning points, rises from Q3 and leaves through Q1. Which family fits?

For parents and teachers

Big idea (C1.4): polynomials, exponentials and sinusoidal functions leave different fingerprints: in polynomials x is the base; in exponentials x is the exponent; in sinusoidal functions x is inside sin or cos. Tables show constant differences, constant ratios, or repeating values.

How to help: Name a real situation and ask which family fits: bacteria doubling every hour (exponential), a Ferris wheel seat's height (sinusoidal), the volume of a box as you change one dimension (polynomial). Ask your teen to explain the reason, not just the name.

Common difficulties:

Confusing x⁴ with 4ˣ.
Expecting differences of exponential data to become constant if you go far enough (they never do).
Calling any wavy graph sinusoidal; a polynomial can wiggle a few times, but its ends always run off to infinity.