# A Quick Guide to Recognizing Linear,Quadratic,and Exponential Functions

# A Quick Guide to Recognizing Linear, Quadratic, and Exponential Functions

**1. Tables**

Important Note: The Input Values MUST be equally spaced

Linear Function |
Quadratic Function |
Exponential Function |

Output difference is constant |
First difference is linear Second difference is constant |
Output ratios are constant |

Important note: There are many different forms of both the
explicit and recursive equations for

the same function.

__Linear:__ Terms are constants or constant times a
variable. Examples: y = mx + b ,

2· (H
-1) = T , y - 3 = 7x + 8, and Output = 2 x Input -1.

__Quadratic:__ Terms include constant times the square
of the input variable and can also include

linear terms as above. Examples: Area = (side)^{2}, y = ax^{2} + bx + c , and

__Exponential:__ A constant times a base raised to a
variable exponent. Examples: y = 2^{x} ,

K = 3·2^{P}
, Balance = 1000 x (1.005)^{months}.

*Copyright 2005, Debra K. Borkovitz. You may copy or
edit this material for non-profit,
educational use only.*

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**3. Recursive Equations
**

__Linear:__Initial Row = ____ (fill in blank with a number); Next Row = Previous Row + ___

(fill in blank with a number )

Example: In the first table, the initial output value is
1, and we add 2 to get the next value.

More formally: f (1) =1, f (n +1) = f (n) + 2

__Quadratic:__ Initial Row = ____ (fill in blank with a
number); Next Row = Previous Row + ___

(fill in blank with a linear function )

Example: In the second table on the other page, the output
value in the initial row is 3, and we

add 4 x (Input) +2 to get the next value. More formally: f (1) = 3, f (n +1) = f
(n) + 4n + 2

__Exponential:__ Initial Row = ____ (fill in blank with
a number); Next Row = Previous Row x ___

(fill in blank with a number )

Example: In the third table on the other page, the output
value in the initial row is 3, and we

multiply by 2 to get the next value. More formally: f (1) = 3, f (n +1) = 2·
f (n) .

**4. Graphs**

**Note 1: You cannot tell for sure whether a function is
quadratic or exponential just from
the graph. There are other functions whose graphs look like quadratics and
exponentials.**

Note 2: Be careful if the domain (possible input values)
is restricted. For example, in many

physical problems, it makes no sense to include negative inputs , but this
restriction takes away

some of the information that might help you identify the shape of the graph.

__Linear: __A straight line

__Quadratic: __A parabola. Has a maximum or a minimum,
and is symmetric about a vertical axis.

Often looks “U Shaped,” but can be deceptive; for example, if small portions are
magnified they

can look like straight lines.

Examples on next page

__Exponential:__ Either grows or decays at a rate
proportional to the function, so eventually either

starts growing very quickly or shrinking to zero very quickly.

Examples on next page

Examples of Quadratic Functions

Examples of Exponential Functions (note the last one
hasn’t “taken off” yet; note similarities with

the third quadratic function).

*Copyright 2005, Debra K. Borkovitz. You may copy or
edit this material for non-profit,
educational use only.*

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