Grade by Grade
D. Characteristics of Functions · Combining Functions and Modelling

Comparing the Function Families

D3.1About 75 minutes
Learning goals
  • I can compare polynomial, rational, exponential, logarithmic and trigonometric functions by their key features.
  • I can identify a function family from its graph, its equation or a table of values (finite differences and ratios).
  • I can compare the long-run growth of exponential, polynomial and logarithmic functions.
  • I can choose a function family that fits a description of a real situation.

Words to know

function family
A group of functions with the same basic equation form and graph shape, such as all exponential functions y = a·bˣ + c.
finite differences
Differences between consecutive y-values in a table with equally spaced x-values. For a polynomial of degree n, the nth differences are constant.
common ratio
The constant you get by dividing each y-value by the one before it. Equally spaced x-values with a common ratio suggest an exponential function.
end behaviour
What happens to y as x → ∞ and as x → −∞.
periodic
Repeating its values over and over at regular intervals, like sine and cosine.

A lineup of suspects

You now know nine kinds of functions: linear, quadratic, cubic and quartic polynomials, rational functions, exponential and logarithmic functions, and the trigonometric functions sine/cosine and tangent. When a scientist, economist or engineer sees new data, the first question is: which family does this belong to?

In this lesson you will compare the families in three ways, just as the curriculum asks: by their graphs (key features), by their equations (algebraic forms), and by their tables of values (finite differences and ratios). Each earlier unit of this course (and Functions, MCR3U, in Grade 11) studied one or two families; now you will put them side by side.

linear: y = 2x − 1quadratic: y = x² − 2cubic: y = x³ − 3xquartic: y = x⁴ − 4x²rational: y = (x + 1)/(x − 1)exponential: y = 2ˣlogarithmic: y = log₂ xsinusoidal: y = sin xtangent: y = tan x
One member of each family. Dashed lines are asymptotes. Look for: turning points (polynomials), asymptotes (rational, exponential, logarithmic, tangent) and repetition (sine and tangent).
FamilyTypical formDomainRangeAsymptotesOther key features
Lineary = mx + bℝℝ (if m ≠ 0)noneconstant rate of change
Quadraticy = a(x − h)² + kℝy ≥ k or y ≤ knoneone turning point; symmetric
Cubic, quartic (degree n)y = a(x − r₁)(x − r₂)…ℝℝ (odd n); bounded on one side (even n)noneup to n zeros, up to n − 1 turning points
Rationaly = (ax + b)/(cx + d), y = 1/f(x)excludes zeros of denominatoroften excludes one valuevertical and horizontalcan have holes
Exponentialy = a·bˣ + cℝy > c (a > 0)horizontal y = cconstant ratio; growth or decay
Logarithmicy = a log_b(x − d) + cx > dℝvertical x = dinverse of exponential; slow growth
Sinusoidaly = a sin(k(x − d)) + cℝc − |a| ≤ y ≤ c + |a|noneperiodic, period 2π/|k|
Tangenty = tan xx ≠ π/2 + nπℝvertical, every πperiodic, period π
Key features of the families studied in MHF4U.

Telling families apart from a table

With equally spaced x-values:

  • Linear: first differences are constant.
  • Quadratic: second differences are constant. Cubic: third differences. In general, a polynomial of degree n has constant nth differences (you met this in the sibling unit Polynomial Functions and Equations).
  • Exponential: the ratio of consecutive y-values is constant. (Its differences are never constant: they grow or shrink by the same ratio.)
  • Sinusoidal: the values repeat in a cycle, rising and falling.

With x-values that multiply by the same number each time (like 1, 2, 4, 8):

  • Logarithmic: the y-values go up by a constant difference. (Multiplying x by 2 adds log_b 2 to y.)

A rational function often shows values that blow up near an x-value (a vertical asymptote) or level off toward a constant (a horizontal asymptote).

Worked example

Example 1: Identify the family from each table

A: x = 0, 1, 2, 3, 4 with y = 0, 1, 14, 51, 124. B: x = 0, 1, 2, 3, 4 with y = 5, 10, 20, 40, 80. C: x = 1, 2, 4, 8, 16 with y = 3, 4, 5, 6, 7. D: x = 0, 1, 2, ..., 8 with y = 0, 2.83, 4, 2.83, 0, −2.83, −4, −2.83, 0.

  1. A: first differences 1, 13, 37, 73; second 12, 24, 36; third 12, 12. Third differences are constant: cubic. (It is y = 2x³ − x; the third difference 12 = 2 × 3!.)
  2. B: differences 5, 10, 20, 40 (not constant), ratios 10/5 = 20/10 = ... = 2. Exponential: y = 5(2)ˣ.
  3. C: x doubles each time while y increases by 1 each time: logarithmic, y = 3 + log₂ x.
  4. D: the values rise to 4, fall to −4 and return to 0 after 8 steps: sinusoidal with amplitude 4 and period 8, y = 4 sin(πx/4).

A cubic, B exponential, C logarithmic, D sinusoidal.

Who wins the race to infinity?

As x → ∞, these families all grow without bound: polynomials (with positive leading coefficient), exponentials with b > 1, and logarithms. But they grow at very different speeds:

logarithmic ≪ polynomial ≪ exponential (in the long run).

A polynomial may be ahead at first, but any exponential function with base b > 1 eventually overtakes any polynomial. And log x grows more slowly than any positive power of x, even √x. You can see the beginning of this race in the graph below; in the sibling unit Rates of Change you saw why: an exponential's rate of change is proportional to its own value, so it speeds up as it grows.

-2-11234551015202530xy(2, 4)(4, 16)y = 2ˣy = x²They meet three times:x ≈ −0.77, x = 2, x = 4.For x > 4 the exponentialstays ahead forever.
2ˣ (solid) and x² (dashed) cross three times. Between x = 2 and x = 4 the parabola is ahead, but from x = 4 onward the exponential wins and keeps pulling away.
Worked example

Example 2: Compare key features

Compare f(x) = 2ˣ, g(x) = x², h(x) = log₂ x and k(x) = 1/x² on: domain, range, symmetry, asymptotes, and behaviour as x → ∞.

  1. Domain: f and g: ℝ. h: x > 0. k: x ≠ 0.
  2. Range: f: y > 0. g: y ≥ 0. h: ℝ. k: y > 0.
  3. Symmetry: g and k are even. f and h are neither.
  4. Asymptotes: f: y = 0 (as x → −∞). h: x = 0. k: x = 0 and y = 0. g: none.
  5. As x → ∞: f → ∞ fastest, g → ∞ more slowly, h → ∞ very slowly, k → 0.

Even though f, g and h all increase to infinity on the right, they differ in domain, range, symmetry, asymptotes and speed. k decays to 0.

Worked example

Example 3: Choose a family for each situation

Which family fits best? (a) the height of the tide in a Bay of Fundy harbour over two days (b) the temperature of a cooling pie (c) the time to drive 100 km as a function of speed (d) the area of a square patio as a function of its side length (e) the sound level in decibels as a function of sound intensity.

  1. (a) Tides rise and fall in a regular cycle: sinusoidal.
  2. (b) The difference from room temperature decreases by the same percentage each minute: exponential (plus a constant).
  3. (c) t = 100/v: rational (time is inversely proportional to speed, with a vertical asymptote at v = 0).
  4. (d) A = s²: quadratic (polynomial).
  5. (e) The decibel scale is defined with a logarithm (sibling unit Exponential and Logarithmic Functions): logarithmic.

(a) sinusoidal (b) exponential (c) rational (d) quadratic (e) logarithmic.

Use more than one clue

A short table can fool you. The points (0, 1), (1, 2), (2, 4) fit y = 2ˣ, but also the quadratic y = 0.5x² + 0.5x + 1. Check more values, think about the context (does it repeat? level off? have a forbidden value?), and look at the shape of the graph before deciding.

Just for fun

Funny but true: x¹⁰ starts out far ahead of 2ˣ (at x = 10 it is ten billion against 1024), yet 2ˣ catches up and passes it for good at about x = 58.8. Exponentials are the tortoise who turns out to have a rocket.

Hands-on activity

Function family card sort

You need: 27 small pieces of paper (or index cards), a pencil and a calculator.

  1. Make 9 equation cards: y = 2x − 1, y = x² − 2, y = x³ − 3x, y = x⁴ − 4x², y = (x + 1)/(x − 1), y = 2ˣ, y = log₂ x, y = sin x, y = tan x.
  2. Make 9 feature cards, one per family, without naming the family, for example "vertical asymptote x = 1 and horizontal asymptote y = 1", "period π, vertical asymptotes", "domain x > 0, passes through (1, 0)", "third differences constant", "range y > 0, ratio 2 between consecutive values", and so on.
  3. Make 9 sketch cards: a quick sketch of each graph (cover the gallery at the top of this lesson first!).
  4. Shuffle all 27 cards and sort them into 9 matching triples. Ask a family member to time you.

What to look for: each triple has one equation, one feature card and one sketch that agree. Check with the gallery and the table in this lesson. The most common mix-ups are cubic with quartic (count turning points and look at the end behaviour) and exponential with quadratic (an exponential has a horizontal asymptote on one side; a parabola does not).

Your turn

1

x = 0, 1, 2, 3, 4 and y = 2, 5, 10, 17, 26. Which family? Find the equation.

2

x = 0, 1, 2, 3 and y = 81, 27, 9, 3. Which family? Find the equation.

3

Which families have a vertical asymptote? Which have a horizontal asymptote?

4

Which families are periodic?

5

Order from slowest to fastest growth as x → ∞: 3ˣ, log x, x⁵, √x.

6

A quantity drops quickly at first and then levels off at 15. Name two families that could model it.

More practice: warm-up

1

Constant first differences: which family?

2

Constant ratios: which family?

3

Which family is the inverse of the exponential family?

4

What is the period of y = sin x?

5

Can a quadratic function have a horizontal asymptote?

More practice: on your own

1

x = 1, 3, 9, 27, 81 and y = 2, 4, 6, 8, 10. Which family? Find the equation.

2

x = −2, −1, 0, 1, 2 and y = −14, 1, 2, 1, −14. Which family? Is it even?

3

Compare y = x³ and y = 3ˣ for x = 0, 1, 2, 3, 4. Which is larger at x = 4? Which wins in the long run?

4

Give the domain, range and asymptotes of y = (2x + 1)/(x − 3) and of y = log(x − 3).

5

The number of hours of daylight in Iqaluit through a year: which family, and why?

Challenge

1

A table with x = 0, 1, 2, 3, 4, 5 has y = 1, 3, 9, 27, 81, 243. Show that no polynomial of degree 4 or less fits all six points exactly, using differences.

2

Find the x-value where 1.5ˣ first passes x² for good (x > 0), using a table of whole numbers.

3

Explain why log x eventually grows more slowly than √x, using the inverse functions 10ˣ and x².

Check your understanding

Quiz

1.

A table with equally spaced x-values has constant second differences. Which family is it?

2.

Which function has a horizontal asymptote but no vertical asymptote?

3.

x = 1, 10, 100, 1000 gives y = 4, 5, 6, 7. Which family fits best?

4.

Which function eventually grows the fastest as x → ∞?

5.

Which family is periodic with vertical asymptotes?

6.

The time needed to drive 300 km depends on the average speed. Which family models it?

For parents and teachers

Big idea: D3.1 asks students to compare the characteristics of polynomial, rational, trigonometric, exponential and logarithmic functions using graphs, algebraic forms and tables of values (finite differences). This lesson is a review and synthesis of the whole course.

How to help: play "Name that family": describe a situation (a bouncing ball's height, a car loan balance, ocean tides) and have your teen name a family and justify it. The card-sort activity works well with two people competing.

Common difficulties:

  • Deciding from too few points (three points fit many families).
  • Using finite differences when the x-values are not equally spaced.
  • Confusing exponential growth with quadratic growth on a short interval.
  • Forgetting that a logarithm has a restricted domain and a vertical asymptote.

In the on-your-own question about a quartic table, there is only one fourth difference, so the shape and symmetry are used to confirm the family; the equation y = 2 − x⁴ is then checked against every point.