B. Exponential Functions · Exponential Functions

Exponential, Linear or Quadratic?

B2.1About 70 minutes

Learning goals

I can use finite differences and ratios to decide whether a table is linear, quadratic or exponential.
I can recognize each family from its graph and its equation.
I can explain several ways to tell f(x) = 2ˣ apart from linear and quadratic functions.
I can write the equation y = a × bˣ from a table with a constant ratio.

Words to know

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

second differences
The differences between consecutive first differences. Constant second differences mean a quadratic relation.

common ratio
The number each y-value is multiplied by to get the next one (for equal steps in x). A constant ratio means an exponential relation.

rate of change
How fast y changes compared with x. Linear: constant. Quadratic: changing at a steady rate. Exponential: proportional to the current value.

Three families, three fingerprints

You now know three families of functions: linear (y = mx + b), quadratic (y = ax² + bx + c) and exponential (y = a × bˣ). Real data rarely arrives with a label. A scientist, an economist or a coach has to look at a table or a graph and decide which family fits. Each family leaves its own fingerprint in tables, graphs and equations.

Remember from Grade 10

In a table with equal steps in x: constant first differences mean linear; constant second differences mean quadratic. In Grade 10 you also saw that y = 2ˣ has neither.

The exponential fingerprint: a constant ratio

Look at y = 2ˣ. The first differences are 1, 2, 4, 8: they repeat the same doubling pattern, so they never become constant, and neither do the second differences (or any later differences). But divide each y-value by the one before it: 2 ÷ 1 = 2, 4 ÷ 2 = 2, 8 ÷ 4 = 2. The ratio is constant.

This is the key test: if the x-values go up in equal steps and the y-values are multiplied by the same number each step, the relation is exponential. Linear functions add the same amount each step; exponential functions multiply by the same amount each step.

For y = 2ˣ the first and second differences keep changing (1, 2, 4, 8 and 1, 2, 4), but every step multiplies y by 2: a constant ratio.

	Linear	Quadratic	Exponential
Equation	y = mx + b	y = ax² + bx + c	y = a × bˣ (b > 0, b ≠ 1)
Where is x?	x to the power 1	highest power x²	x is the exponent
Table (equal x-steps)	1st differences constant	2nd differences constant	ratio of y-values constant
Rate of change	constant	changes steadily	proportional to y: grows as y grows
Graph	straight line	parabola with a vertex and axis of symmetry	curve with a horizontal asymptote; no vertex
Intercepts	can cross the x-axis once	0, 1 or 2 x-intercepts	y = a × bˣ has no x-intercept

Comparing linear, quadratic and exponential functions.

The three shapes. A line keeps one slope. A parabola turns around at its vertex and is symmetric. An exponential curve never turns around and levels off toward an asymptote on one side.

Worked example

Example 1: Classify each table

Each table uses x = 0, 1, 2, 3, 4. Decide whether each relation is linear, quadratic or exponential.

A: y = 5, 8, 11, 14, 17

B: y = 2, 6, 18, 54, 162

C: y = 1, 3, 9, 19, 33

D: y = 64, 48, 36, 27, 20.25

A: first differences 3, 3, 3, 3: constant → linear (y = 3x + 5).
B: first differences 4, 12, 36, 108 (not constant). Ratios 6 ÷ 2 = 3, 18 ÷ 6 = 3, 54 ÷ 18 = 3, 162 ÷ 54 = 3: constant → exponential (y = 2 × 3ˣ).
C: first differences 2, 6, 10, 14; second differences 4, 4, 4: constant → quadratic (y = 2x² + 1). Ratios 3, 3, 2.1, 1.7 are not constant.
D: first differences −16, −12, −9, −6.75 (not constant). Ratios 48 ÷ 64 = 0.75, 36 ÷ 48 = 0.75, 27 ÷ 36 = 0.75, 20.25 ÷ 27 = 0.75: constant → exponential decay (y = 64 × 0.75ˣ).

A linear, B exponential, C quadratic, D exponential.

Worked example

Example 2: Classify from the equation

Classify: (a) y = 4x² − 1 (b) y = 4ˣ − 1 (c) y = 4x − 1 (d) y = 4(0.5)ˣ (e) y = x⁴

Ask: where is the variable?
(a) highest power x²: quadratic.
(b) x is the exponent: exponential (shifted down 1, as you will see in Lesson 8).
(c) x to the power 1: linear.
(d) x is the exponent and the base 0.5 is between 0 and 1: exponential decay.
(e) The variable is the base with a fixed exponent 4: a polynomial (power) function, not exponential.

(a) quadratic, (b) exponential, (c) linear, (d) exponential, (e) neither (polynomial).

Worked example

Example 3: Several ways to tell f(x) = 2ˣ apart

Give three ways to tell f(x) = 2ˣ from the quadratic g(x) = x² and the linear h(x) = 2x.

Tables: for x = 0, 1, 2, 3, 4: f gives 1, 2, 4, 8, 16 (ratio 2); g gives 0, 1, 4, 9, 16 (second differences 2); h gives 0, 2, 4, 6, 8 (first differences 2).
Graphs: h is a straight line; g is a parabola with vertex (0, 0) and symmetric about the y-axis; f has no vertex, is not symmetric and has the asymptote y = 0.
Equations and values: f has the variable in the exponent; f(x) is always positive, while g(0) = 0 and h(x) < 0 for x < 0.
Long-run behaviour: eventually f grows faster than both. At x = 10: f = 1024, g = 100, h = 20.

Use constant ratio vs constant differences, the shape of the graph (asymptote vs vertex vs line), and where the variable sits in the equation.

Careful

The difference and ratio tests only work when the x-values go up in equal steps. If the x-steps are uneven, either fill in more of the table or compare the graph and equation instead. Also, real data is rarely perfect: ratios of 2.9, 3.1, 3.0, 2.95 suggest an exponential model with a ratio of about 3.

Just for fun

A linear function, a quadratic function and an exponential function enter a 100-step race. Linear sets a nice steady pace. Quadratic slowly speeds up and passes it.

Exponential (y = 2ˣ) starts out slowly and waves politely... and by step 30 it is more than a billion steps ahead. The others are still looking for it.

Hands-on activity

Which allowance would you choose?

You need: paper, a pencil and a calculator.

A (very generous) relative offers you a choice of three allowance plans for the next 15 days:

Plan L: $5 on day 1, and $5 more each day than the day before ($5, $10, $15, …).
Plan Q: $1 × (day number)² ($1, $4, $9, …).
Plan E: 1 cent on day 1, doubling every day (1¢, 2¢, 4¢, …).

Make a table of the amount received each day for each plan, for days 1 to 15.
Find the first differences, second differences and ratios for each plan. Label each plan linear, quadratic or exponential.
On which days does each plan pay more than the other two? Which plan pays the most on day 15?
Sketch all three on one graph.

What to look for: Plan L has constant first differences of $5 (linear). Plan Q has constant second differences of $2 (quadratic). Plan E has a constant ratio of 2 (exponential): its day amount is 0.01 × 2ⁿ⁻¹ dollars. Plan L pays the most on days 1 to 4, L and Q tie on day 5 ($25 each), and Q pays the most on days 6 to 15. On day 15: L pays $75, Q pays $225, E pays 0.01 × 2¹⁴ = $163.84. Plan E is behind for most of the two weeks, but its day-16 payment ($327.68) would already beat Q's ($256), and it pulls away fast after that.

Your turn

x = 0, 1, 2, 3; y = 3, 12, 48, 192. Linear, quadratic or exponential? Find the equation.

x = 0, 1, 2, 3; y = 10, 7, 4, 1. Classify and find the equation.

x = 0, 1, 2, 3, 4; y = 0, 2, 8, 18, 32. Classify.

x = 0, 1, 2, 3; y = 100, 50, 25, 12.5. Classify and find the equation.

Classify y = 3(1.2)ˣ, y = 1.2x³ and y = 1.2x + 3.

Give two features of the graph of y = 5ˣ that the graph of y = x² does not have.

More practice: warm-up

What do you check to see if a table is exponential?

Constant first differences mean the relation is…

Constant second differences mean the relation is…

y = 7ˣ: where is the variable?

Which graph has an asymptote: a line, a parabola or y = 3ˣ?

More practice: on your own

x = 1, 2, 3, 4; y = 6, 18, 54, 162. Find an exponential equation. (Careful: the table does not start at x = 0.)

x = 0, 2, 4, 6; y = 1, 4, 16, 64. Is it exponential? What is y when x = 1?

Data from an experiment: x = 0, 1, 2, 3, 4; y = 80, 61, 45, 34, 26. Which model fits best? Explain.

Show that the first differences of y = 3ˣ (x = 0, 1, 2, 3, 4) also have a constant ratio.

Which grows faster for large x, y = 100x² or y = 1.5ˣ? Test x = 30.

Challenge

Is the relation with x = 0, 1, 2, 3, 4 and y = 2, 5, 11, 23, 47 linear, quadratic or exponential? (Hint: look at the first differences.)

A table has equal x-steps and constant ratio 1. What kind of function is it, and why is the base 1 excluded from exponential functions?

Prove that for y = a × bˣ, the ratio y(x + 1) ÷ y(x) always equals b.

Check your understanding

Quiz

A table with equal x-steps has a constant ratio of 1.5. The relation is:

x = 0, 1, 2, 3 and y = 4, 7, 12, 19. The relation is:

Which equation is exponential?

Which graph has a horizontal asymptote?

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

The difference and ratio tests need:

For parents and teachers

Big idea: Expectation B2.1: distinguish exponential functions from linear and quadratic functions using finite differences, constant ratios, graphs and equations.

How to help: Give your teen a quick table (for example 5, 10, 20, 40 or 5, 10, 15, 20) and ask "add or multiply?" Linear adds the same amount; exponential multiplies by the same amount.

Common difficulties:

Using differences or ratios when the x-steps are not equal.
Stopping at "the first differences aren't constant, so it's quadratic". Check the second differences and the ratios.
Confusing y = x³ (polynomial) with y = 3ˣ (exponential).
Expecting perfect ratios in real data. Nearly constant ratios still point to an exponential model.