Before you read

Answer first — see what you already know.

A city's population increases by $8%$ each year. Which function type best models this?

A
Linear
B
Quadratic
C
Exponential
D
Polynomial (degree 3)

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A data table shows $y$ values $5, 8, 13, 20$ for $x = 0, 1, 2, 3$ . What type of function fits?

A
Linear
B
Quadratic
C
Exponential
D
None of these

Check your answer

Answer:

Theory

The Three Main Function Types

Property	Linear	Quadratic	Exponential
Equation	$f(x) = mx + b$	$f(x) = ax^2 + bx + c$	$f(x) = a \cdot b^x$
Graph	Straight line	Parabola	J-curve
Growth	Constant rate (add)	Accelerating (add increasing amounts)	Multiplicative (multiply)
Table test	Constant 1st differences	Constant 2nd differences	Constant ratios
Key feature	Slope	Vertex	Asymptote

Linear: constant rate of change

Exponential: increasing rate of change

Theory

How to Tell Them Apart from a Table

Given equally spaced $x$ -values:

1st differences = differences between consecutive $y$ -values.
2nd differences = differences between consecutive 1st differences.
Ratios = each $y$ -value divided by the previous one.

Pattern	Function type
Constant 1st differences	Linear
Constant 2nd differences	Quadratic
Constant ratios	Exponential

Example 1

Identify the type:

$x$	$y$
0	3
1	7
2	11
3	15

1st differences: $7-3=4$ , $11-7=4$ , $15-11=4$ . Constant!

Linear (constant rate of change $= 4$ ).

Linear

Example 2

Identify the type:

$x$	$y$
0	1
1	3
2	9
3	27

1st differences: $2, 6, 18$ — not constant.

Ratios: $3/1=3$ , $9/3=3$ , $27/9=3$ . Constant!

Exponential (common ratio $= 3$ ).

Exponential

Example 3

Identify the type:

$x$	$y$
0	1
1	4
2	9
3	16

1st differences: $3, 5, 7$ — not constant.

2nd differences: $5-3=2$ , $7-5=2$ . Constant!

Quadratic (these are perfect squares: $y = (x+1)^2$ ).

Quadratic

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