B. Exponential Functions · Exponential Functions

Linear, Quadratic or Exponential? Comparing Three Kinds of Functions

B1.6About 70 minutes

Learning goals

I can tell linear, quadratic and exponential functions apart using tables (first differences, second differences, ratios).
I can tell them apart from their graphs and their equations.
I can compare how the rate of change behaves in each kind of function.
I can compare all three in the same situation and explain why exponential growth eventually overtakes the others.

Words to know

first differences
The differences between consecutive y-values in a table with equal x-steps. Constant first differences mean linear.

second differences
The differences between consecutive first differences. Constant (non-zero) second differences mean quadratic.

ratio (common ratio)
Each y-value divided by the one before it. A constant ratio (not 1) means exponential.

rate of change
How fast y changes compared with x. Constant for linear functions, changing steadily for quadratics, and proportional to the current amount for exponentials.

parabola
The U-shaped graph of a quadratic function. It has a vertex and an axis of symmetry.

Three predictions for one club

A new school robotics club starts with 100 followers on its page. Three students each predict how the number will grow, w weeks from now:

Linear: gain 60 followers every week: L(w) = 100 + 60w
Quadratic: the gains get bigger each week: Q(w) = 100 + 30w²
Exponential: grow by 50% every week: E(w) = 100(1.5)ʷ

At first the three predictions are very close. After two months they are wildly different. Being able to tell which kind of function fits a situation is one of the most useful skills in this course.

Week w	0	1	2	3	4	5	6
L(w) = 100 + 60w	100	160	220	280	340	400	460
Q(w) = 100 + 30w²	100	130	220	370	580	850	1180
E(w) = 100(1.5)ʷ	100	150	225	337.5	506.25	759.38	1139.06

The three predictions for the first six weeks (E rounded to 2 decimal places).

Test 1: Tables

With equal steps in x, look at the y-values three ways:

Linear: first differences are constant. L: +60, +60, +60, …
Quadratic: first differences change, but second differences are constant. Q: first differences 30, 90, 150, 210, 270, 330; second differences 60, 60, 60, …
Exponential: differences are not constant, but the ratios are. E: 150/100 = 1.5, 225/150 = 1.5, 337.5/225 = 1.5, …

A neat fact: the first differences of an exponential table (50, 75, 112.5, 168.75, …) are themselves exponential, with the same ratio 1.5. So the second, third, … differences never become constant.

Checking a table with equal x-steps. If none of the three tests works, the relation is none of these three types (or the data are messy and only roughly fit one).

Test 2: Graphs

Linear: a straight line.
Quadratic: a parabola, which has a vertex and is symmetric; it turns around.
Exponential: a curve that never turns around, has a horizontal asymptote on one side and grows (or decays) faster and faster on the other.

Test 3: Equations

Linear: y = mx + b (x to the power 1).
Quadratic: y = ax² + bx + c, a ≠ 0 (highest power of x is 2).
Exponential: y = a(b)ˣ, b > 0, b ≠ 1 (x is in the exponent).

Rate of change

A linear function changes by the same amount each step. A quadratic's change grows by the same amount each step. An exponential function changes by the same percent each step, so its change is proportional to how big it already is. The bigger it gets, the faster it grows.

The three predictions for 8 weeks. The quadratic prediction leads in weeks 3 to 6, but by week 7 the exponential one has passed it, and then it leaves both others far behind.

Worked example

Example 1: Classify from tables

Each table uses x = 0, 1, 2, 3, 4. Classify each relation.

A: y = 3, 7, 11, 15, 19 B: y = 2, 6, 18, 54, 162 C: y = 1, 2, 5, 10, 17 D: y = 64, 48, 36, 27, 20.25

A: first differences 4, 4, 4, 4 are constant: linear.
B: first differences 4, 12, 36, 108 are not constant. Ratios 6/2 = 18/6 = 54/18 = 162/54 = 3: exponential (growth).
C: first differences 1, 3, 5, 7; second differences 2, 2, 2: quadratic.
D: first differences −16, −12, −9, −6.75 are not constant; second differences 4, 3, 2.25 are not constant. Ratios 48/64 = 0.75, 36/48 = 0.75, 27/36 = 0.75, 20.25/27 = 0.75: exponential (decay).

A linear, B exponential, C quadratic, D exponential

Worked example

Example 2: Classify from equations

Classify: (a) y = 4x − 9 (b) y = 4(0.9)ˣ (c) y = 4x² − 9 (d) y = 2(x − 1)² + 5 (e) y = 3ˣ + 2

(a) x has power 1: linear.
(b) x is in the exponent: exponential (decay, since 0.9 < 1).
(c) highest power of x is 2: quadratic.
(d) vertex form of a parabola: quadratic.
(e) x is in the exponent: exponential, with asymptote y = 2.

linear, exponential, quadratic, quadratic, exponential

Worked example

Example 3: Compare in the same context

Using the club predictions, (a) in which weeks is the quadratic prediction the largest? (b) When does the exponential prediction become the largest for good? (c) Find all three predictions for week 10 and comment.

(a) From the table: week 2 has E = 225 > Q = 220, but weeks 3 to 6 have Q > E > L (for example week 4: 580 > 506.25 > 340). Q is largest in weeks 3 to 6.
(b) Week 7: Q = 100 + 30(49) = 1570, E = 100(1.5)⁷ ≈ 1708.6. From week 7 on, E is the largest, and the gap keeps widening because E multiplies while Q only adds.
(c) L(10) = 700; Q(10) = 3100; E(10) = 100(1.5)¹⁰ ≈ 5767. The exponential prediction is almost double the quadratic one and more than 8 times the linear one.

(a) weeks 3–6 (b) from week 7 onward (c) 700, 3100, about 5767

Messy real data

Real data never fit perfectly. Look for differences or ratios that are roughly constant. If the ratios are all close to 1.2 but the first differences keep growing, an exponential model is the better choice. Graphing technology (Desmos, GeoGebra or a graphing calculator) can fit each type of model and show which curve stays closest to the points.

Just for fun

A linear function, a quadratic function and an exponential function enter a race.

The linear one keeps a steady pace, the quadratic one speeds up, and the exponential one starts slowly, so the others laugh. Then the exponential function doubles its speed. And doubles again. By the finish line, the others are still looking for the starting line on the graph.

Hands-on activity

Three ways to stack coins

You need: a pile of coins (or buttons or dried beans, about 100), and paper.

Linear: make piles where each pile has 2 more coins than the last, starting with 1: 1, 3, 5, … Make 5 piles and record the counts.
Quadratic: make square arrangements with sides 1, 2, 3, 4, 5. Record the number of coins in each (1, 4, …).
Exponential: make piles that double, starting with 1: 1, 2, 4, … Make as many as your coins allow.
For each list find the first differences, second differences and ratios. Which test works for which list?
How many piles could you make in step 3 with 100 coins in total? With 1000?

What to look for: linear 1, 3, 5, 7, 9 (first differences all 2). Quadratic 1, 4, 9, 16, 25 (second differences all 2). Exponential 1, 2, 4, 8, 16, 32 (ratios all 2). With 100 coins, six doubling piles use 1 + 2 + 4 + 8 + 16 + 32 = 63 coins, and a seventh pile of 64 would need 127 in all, so only 6 complete piles fit. Even 1000 coins only allow 9 piles (1 + 2 + … + 256 = 511; adding 512 needs 1023). Doubling uses up any supply very quickly.

Your turn

Classify (x = 0, 1, 2, 3): y = 5, 10, 20, 40.

Classify (x = 0, 1, 2, 3): y = 5, 10, 15, 20.

Classify (x = 0, 1, 2, 3, 4): y = 0, 3, 12, 27, 48.

Classify: y = 7(1.03)ˣ.

Classify: y = −2x² + 8x.

Describe how the graph of an exponential function differs from a parabola.

y = 1000(0.8)ˣ and y = 1000 − 200x both start at 1000. Which reaches 0 first? Explain.

More practice: warm-up

Which test identifies an exponential table?

Which test identifies a quadratic table?

Classify: y = 9x + 2.

Classify: y = 9ˣ.

Classify: y = x⁹.

More practice: on your own

Classify (x = 0, 1, 2, 3, 4): y = 81, 54, 36, 24, 16. Write an equation.

Classify (x = 1, 2, 3, 4, 5): y = 2, 8, 18, 32, 50.

Two phone plans: Plan A costs $30 + $5 per GB. Plan B's price would double every year starting at $5 (a silly plan!). What kind of function is each, in terms of GB or years?

Find the first differences of y = 3(2)ˣ for x = 0 to 5. What do you notice?

In the club example, find the week when the exponential prediction first passes 1000 followers. Which of the other predictions is bigger that week?

Challenge

y = 2ˣ and y = x² are equal at x = 2 and x = 4. Which is bigger at x = 3, and at x = 10? What happens for x > 4?

A table has x = 0, 2, 4, 6 (steps of 2) and y = 10, 40, 160, 640. Is it exponential? Find the equation.

Create one table of five values (x = 0 to 4) that starts at y = 8 for each type: linear, quadratic (not linear), and exponential. Show the test that proves each type.

Check your understanding

Quiz

A table with equal x-steps has y-values 4, 12, 36, 108. What kind of relation is it?

Which equation is quadratic?

The y-values 3, 5, 9, 15, 23 (equal x-steps) are best described as:

Which graph never turns around and has a horizontal asymptote?

Which statement about the rate of change is true?

f(x) = 100 + 50x and g(x) = 10(2)ˣ. Which is true?

For parents and teachers

Big idea (B1.6): students distinguish exponential functions from linear and quadratic functions by comparing tables (finite differences and ratios), graphs, equations and rates of change, ideally in the same context.

How to help: Make up a quick table together (a savings plan, a growing snowball fight team, a pile of coins) and ask, "Is it adding the same amount, or multiplying by the same number?" That question alone separates most linear and exponential situations.

Common difficulties:

Calling every curve exponential: a parabola curves too, but it turns around and has constant second differences.
Using unequal x-steps: the difference and ratio tests only work with equal steps in x.
Checking only one or two ratios: check them all; one ratio means nothing.
Thinking a big early lead means a function stays ahead: exponential growth (base > 1) always overtakes linear and quadratic growth eventually.