Comparing Function Types

You’ve now met five big families of functions: polynomial, rational, exponential, logarithmic, and trigonometric (sinusoidal). Each has its own shape, its own equation, and its own “fingerprint” in a table of values. Being able to tell them apart quickly is what lets you pick the right model for real data and the right method for solving an equation. Trig functions here use radians.

Key ideas

The families at a glance

Typical graphs of each function type. Dashed lines are asymptotes.

Type	Typical equation	Domain	Asymptotes?	Special features
Polynomial	$a_n x^n + \dots + a_1 x + a_0$	all reals	none	up to $n$ zeros, up to $n - 1$ turning points; end behaviour set by degree and leading coefficient
Rational	$\dfrac{p(x)}{q(x)}$	all reals except zeros of $q$	vertical (or holes), often horizontal	separate branches
Exponential	$a \cdot b^{k(x - d)} + c$	all reals	horizontal, $y = c$	always increasing or always decreasing; no zeros when $c = 0$
Logarithmic	$a \log_b\big(k(x - d)\big) + c$	one side of $x = d$	vertical, $x = d$	always increasing or always decreasing; grows very slowly
Sinusoidal	$a\sin\big(k(x - d)\big) + c$	all reals	none	periodic, range $c - \lvert a \rvert \le y \le c + \lvert a \rvert$

Symmetry

• Polynomials with only even powers (like $x^4 - 3x^2$) are even; with only odd powers (like $x^3 - x$), odd. See even and odd functions.
• $\sin x$ is odd and $\cos x$ is even.
• Exponential and logarithmic functions are never even or odd: their graphs are lopsided.

Fingerprints in a table

When the $x$-values go up in equal steps:

• Linear: the first differences are constant.
• Polynomial of degree $n$: the $n$th differences are constant. For steps of $1$, the $n$th difference equals $a \cdot n!$, where $a$ is the leading coefficient (for a quadratic, the second difference is $2a$).
• Exponential: the ratios of consecutive $y$-values are constant. (The differences are never constant; they grow or shrink by the same ratio.)
• Sinusoidal: the $y$-values repeat in a regular cycle.

When the $x$-values are multiplied by a constant each step (like $1, 10, 100, 1000$):

• Logarithmic: the $y$-values go up by a constant difference.

Rational functions don’t have a neat table fingerprint, but you’ll see values blow up near a vertical asymptote and level off toward a horizontal one.

Rates of change

A linear function changes at a constant rate. A quadratic’s rate of change changes steadily. An exponential function’s rate of change is proportional to its value, so it speeds up (or slows down) faster and faster. A logarithmic function keeps increasing but more and more slowly. A sinusoidal function’s rate of change goes back and forth between positive and negative.

In the long run, any increasing exponential eventually beats any polynomial, and any polynomial eventually beats any logarithm, even if it doesn’t look that way at first (Example 4).

Worked examples

Example 1: Identifying types from tables

Identify the type of function that fits each table, and find an equation.

$x$	$0$	$1$	$2$	$3$	$4$	$5$
A	$3$	$6$	$12$	$24$	$48$	$96$
B	$2$	$5$	$10$	$17$	$26$	$37$
C	$0$	$1$	$0$	$-1$	$0$	$1$
D	$4$	$7$	$10$	$13$	$16$	$19$

Solution. The $x$-values go up in steps of $1$, so look at differences and ratios.

A. Differences: $3, 6, 12, 24, 48$, not constant. Ratios: $\tfrac{6}{3} = \tfrac{12}{6} = \dots = 2$, constant. Exponential, starting at $3$ and doubling: $y = 3(2)^x$.

B. First differences: $3, 5, 7, 9, 11$. Second differences: $2, 2, 2, 2$, constant. Quadratic, with $2a = 2$, so $a = 1$. The $y$-intercept is $2$, so $y = x^2 + bx + 2$. Using $x = 1$: $1 + b + 2 = 5$, so $b = 2$ and $y = x^2 + 2x + 2$. Check $x = 3$: $9 + 6 + 2 = 17$. ✓

C. The values repeat $0, 1, 0, -1$ every $4$ units. Sinusoidal with period $4$, amplitude $1$, axis $y = 0$, starting upward from $0$ like a sine: $k = \tfrac{2\pi}{4} = \tfrac{\pi}{2}$, so $y = \sin\left(\tfrac{\pi}{2}x\right)$. Check $x = 3$: $\sin\tfrac{3\pi}{2} = -1$. ✓

D. First differences: $3, 3, 3, 3, 3$, constant. Linear: $y = 3x + 4$.

Example 2: A logarithmic table

Identify the type of function and find an equation.

$x$	$0.1$	$1$	$10$	$100$	$1000$
$y$	$-1$	$2$	$5$	$8$	$11$

Solution. The $x$-values are multiplied by $10$ each step, while the $y$-values go up by $3$ each step. That’s the fingerprint of a logarithm: multiplying the input by $10$ adds $1$ to $\log x$, so here it adds $3 \times 1$.

Try $y = a\log x + c$. At $x = 1$, $\log 1 = 0$, so $c = 2$. At $x = 10$: $a(1) + 2 = 5$, so $a = 3$.

$$y = 3\log x + 2$$

Check $x = 0.1$: $3(-1) + 2 = -1$. ✓ Check $x = 1000$: $3(3) + 2 = 11$. ✓

(If you’d plotted these points with equally spaced $x$-values, you’d see the slow, flattening curve of a logarithm.)

Example 3: Identifying types from key features

Which type of function (polynomial, rational, exponential, logarithmic, or sinusoidal) could have each set of features?

• (a) Domain all real numbers, range $\{y \in \mathbb{R} \mid y \gt -2\}$, and a horizontal asymptote.
• (b) Domain $\{x \in \mathbb{R} \mid x \gt 3\}$, range all real numbers.
• (c) Range $\{y \in \mathbb{R} \mid -1 \le y \le 5\}$, and the graph repeats every $\pi$ units.
• (d) Vertical asymptote $x = 2$ and horizontal asymptote $y = 1$.
• (e) Domain and range all real numbers, three zeros, and no asymptotes.

Solution.

(a) Exponential: a range bounded on one side, with a horizontal asymptote $y = -2$ and every real number as input. For example, $y = 2^x - 2$.

(b) Logarithmic: the domain stops at a vertical asymptote $x = 3$, but the outputs take every real value. For example, $y = \log(x - 3)$.

(c) Sinusoidal: periodic and bounded, with amplitude $\tfrac{5 - (-1)}{2} = 3$ and axis $y = 2$. For example, $y = 3\sin(2x) + 2$, which has period $\tfrac{2\pi}{2} = \pi$.

(d) Rational: both a vertical and a horizontal asymptote. For example, $y = \dfrac{1}{x - 2} + 1$.

(e) Polynomial, of odd degree (since the range is all real numbers), at least $3$. For example, $y = x^3 - x$, with zeros $-1$, $0$ and $1$.

Example 4: Exponential versus polynomial growth

Compare $f(x) = 2^x$ and $g(x) = x^3$ for $x = 2, 5, 9, 10, 15$. Which function is larger in the long run?

Solution.

$x$	$2$	$5$	$9$	$10$	$15$
$2^x$	$4$	$32$	$512$	$1024$	$32\,768$
$x^3$	$8$	$125$	$729$	$1000$	$3375$

For a while, $x^3$ is ahead. But $2^x$ passes it between $x = 9$ and $x = 10$, and after that it pulls away fast: at $x = 15$ it’s almost ten times bigger. The reason is the rate of change. Each step of $1$ doubles $2^x$, while it multiplies $x^3$ by $\left(\tfrac{x + 1}{x}\right)^3$, which gets closer and closer to $1$. So the exponential wins in the long run, as every increasing exponential does against every polynomial.

The lesson for reading tables: a short table can be misleading. Look at the pattern (ratios versus differences), not just which numbers are bigger right now.

Common mistakes

Mixing up differences and ratios. Constant differences mean linear (or, at a deeper level, polynomial). Constant ratios mean exponential. Check both before deciding.

Using finite differences when the x-steps aren’t equal. Differences only work when $x$ goes up by the same amount each time. Rearrange or re-space the data first, or check for a logarithmic pattern if $x$ is being multiplied.

Thinking every curve that bends is a parabola. Exponential, logarithmic and higher-degree polynomial graphs bend too. Look for asymptotes, symmetry and table patterns, not just the curve.

Expecting zeros from an exponential. $y = a \cdot b^x$ is never $0$. Only a shifted exponential like $y = 2^x - 2$ crosses the $x$-axis.

Deciding from a few values. In Example 4, $x^3$ beats $2^x$ for moderate $x$ (from about $2$ to $9$), but not for long. Think about the long-run behaviour of each family.

Forgetting the restricted domain of a logarithm. A function with a domain like $x \gt 3$ is a strong hint that it’s logarithmic. Polynomials, exponentials and sinusoids accept every real number.

Practice

1. (Warm-up) Identify the type of each function.

• (a) $y = 3(0.5)^x$
• (b) $y = 2x^3 - x$
• (c) $y = \dfrac{x + 1}{x - 2}$
• (d) $y = \log(x - 4)$
• (e) $y = 4\sin(2x) + 1$

Solution

(a) Exponential (decay, since $0 \lt 0.5 \lt 1$).

(b) Polynomial (cubic).

(c) Rational.

(d) Logarithmic.

(e) Sinusoidal.

2. (Warm-up) Use finite differences to identify the type of function, then find its equation.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$-1$	$1$	$7$	$17$	$31$

Solution

First differences: $2, 6, 10, 14$. Second differences: $4, 4, 4$. Constant second differences mean quadratic, with $2a = 4$, so $a = 2$.

The $y$-intercept is $-1$, so $y = 2x^2 + bx - 1$. Using $x = 1$: $2 + b - 1 = 1$, so $b = 0$.

$$y = 2x^2 - 1$$

Check $x = 4$: $32 - 1 = 31$. ✓

3. (Warm-up) From the five types (polynomial, rational, exponential, logarithmic, sinusoidal), which can have:

• (a) a horizontal asymptote?
• (b) a vertical asymptote?
• (c) a graph that repeats?

Solution

(a) Rational and exponential.

(b) Rational and logarithmic.

(c) Sinusoidal.

4. (Core) Identify the type of function and find an equation.

$x$	$0$	$1$	$2$	$3$	$4$
$y$	$80$	$60$	$45$	$33.75$	$25.3125$

Solution

Differences: $-20, -15, -11.25, -8.4375$, not constant. Ratios: $\tfrac{60}{80} = \tfrac{45}{60} = \tfrac{33.75}{45} = \tfrac{25.3125}{33.75} = 0.75$, constant. Exponential decay:

$$y = 80(0.75)^x$$

5. (Core) Find the degree and the leading coefficient of the polynomial function that fits this table.

$x$	$-2$	$-1$	$0$	$1$	$2$	$3$
$y$	$-6$	$0$	$0$	$0$	$6$	$24$

Solution

First differences: $6, 0, 0, 6, 18$. Second: $-6, 0, 6, 12$. Third: $6, 6, 6$.

The third differences are constant, so the degree is $3$. For steps of $1$, the third difference is $a \cdot 3! = 6a$, so $6a = 6$ and $a = 1$.

(The function is $y = x^3 - x$: its zeros $-1$, $0$, $1$ match the table.)

6. (Core) Match each description with a type of function, and give a possible equation.

• (a) Increasing for all $x$, with no zeros, and approaching $0$ as $x$ decreases.
• (b) Decreasing for all $x$ in its domain $\{x \in \mathbb{R} \mid x \gt 0\}$.
• (c) An even function with range $\{y \in \mathbb{R} \mid y \ge -3\}$ and two zeros.

Solution

(a) Exponential growth, for example $y = 2^x$.

(b) Logarithmic with a reflection, for example $y = -\log x$. (A rational function like $y = \tfrac{1}{x}$ restricted to $x \gt 0$ also works, but its natural domain includes negative numbers.)

(c) Quadratic, for example $y = x^2 - 3$, which is even, has minimum $-3$, and has zeros $\pm\sqrt{3}$.

7. (Core) Compare $f(x) = x^2$ and $g(x) = 2^x$ for $x \ge 0$.

• (a) Make a table for $x = 0, 1, 2, 3, 4, 5$.
• (b) For which values of $x \ge 0$ is $x^2 \gt 2^x$?

Solution

(a)

$x$	$0$	$1$	$2$	$3$	$4$	$5$
$x^2$	$0$	$1$	$4$	$9$	$16$	$25$
$2^x$	$1$	$2$	$4$	$8$	$16$	$32$

(b) They’re equal at $x = 2$ and $x = 4$. Between them, $x^2$ is bigger (for example, $9 \gt 8$ at $x = 3$). Before $x = 2$ and after $x = 4$, $2^x$ is bigger, and since the exponential doubles with each step, it stays ahead for good after $x = 4$. So for $x \ge 0$, $x^2 \gt 2^x$ when $2 \lt x \lt 4$.

(Graphing both confirms this. For negative $x$ there’s one more crossing near $x \approx -0.77$.)

8. (Challenge) Identify the type of function and find an equation.

$x$	$2$	$4$	$8$	$16$	$32$
$y$	$1$	$3$	$5$	$7$	$9$

Solution

$x$ doubles each step, and $y$ goes up by $2$ each step. That’s a logarithmic pattern, base $2$: doubling $x$ adds $1$ to $\log_2 x$, so here it adds $2 \times 1$.

Try $y = a\log_2 x + c$. At $x = 2$: $a(1) + c = 1$. At $x = 4$: $a(2) + c = 3$. Subtracting gives $a = 2$, and then $c = -1$.

$$y = 2\log_2 x - 1$$

Check $x = 32$: $2(5) - 1 = 9$. ✓

9. (Challenge) Identify the type of function and find an equation, with $x$ in radians.

$x$	$0$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$
$y$	$3$	$5$	$3$	$1$	$3$	$5$	$3$	$1$	$3$

Solution

The values repeat every $4$ units: sinusoidal. Maximum $5$ and minimum $1$, so the amplitude is $\tfrac{5 - 1}{2} = 2$ and the axis is $y = \tfrac{5 + 1}{2} = 3$. The period is $4$, so $k = \tfrac{2\pi}{4} = \tfrac{\pi}{2}$. At $x = 0$ the graph is on the axis heading up, like a sine:

$$y = 2\sin\left(\frac{\pi}{2}x\right) + 3$$

Check $x = 3$: $2\sin\tfrac{3\pi}{2} + 3 = -2 + 3 = 1$. ✓