Functions Cheat Sheet Name __________________

Exponents are shorthand for repeated multiplication of the same thing by itself.

For instance, the shorthand for multiplying three copies of the number 5 is

(5)(5)(5) = 53.

The "exponent", being 3 in this example, stands for however many times the

value is being multiplied. The thing that's being multiplied, being 5 in this

example, is called the "base".

Zero Exponents

𝑎0 = 1

Product Property of Exponents

𝑎𝑚 ∙ 𝑎𝑛 = 𝑎𝑚+𝑛

Power of a Quotient Property

(𝑎

𝑏)

𝑛

=𝑎𝑛

𝑏𝑛

Definition of Negative Exponents

𝑎−𝑛 =1

𝑎𝑛 or (

𝑎

𝑏)

−𝑛

= (𝑏

𝑎)

𝑛

Power of a Power Property

(𝑎𝑚)𝑛 = 𝑎𝑚𝑛

Power of a Product Property

(𝑎𝑏)𝑚 = 𝑎𝑚𝑏𝑚

Common Base Property of Equality

If 𝑎𝑛 = 𝑎𝑚, and 𝑎 ≠ 1, then 𝑛 = 𝑚.

Quotient Property of Exponents

𝑎𝑚

𝑎𝑛= 𝑎𝑚−𝑛

Power Property of Equality

If 𝑎 = 𝑏, then 𝑎𝑛 = 𝑏𝑛.

Equivalence of Radicals and Rational

Exponents

𝑎1𝑛 = √𝑎

𝑛 and 𝑎

𝑚𝑛 = √𝑎𝑚𝑛

= ( √𝑎𝑛

)𝑚

Scientific Notation

Scientific notation is the way that scientists easily handle very large numbers

or very small numbers.

For example, instead of writing 0.0000000056, we write 5.6 x 10−9.

We can think of 5.6 x 10−9 as the product of two numbers: 5.6 (the digit term)

and 10−9 (the exponential term).

To figure out the power of 10, think "how many places do I move the decimal point?"

When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is

positive.

When the number is smaller than 1, the decimal point has to move to the right, so the power of 10

is negative.

Example: 3 × 10^4 is the same as 3 × 104

3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Example: 0.0055 is written 5.5 × 10−3

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10−3

Scientific Notation in a calculator:

Instead of using “× 10^” your calculator will use “E” to represent scientific

notation.

If you look at the table, when 𝑥 = 4, 𝑌1 = 6𝐸16

This means 6 × 1016

**THESE ARE ROUNDED VALUES**

When we try to calculate 248968, we get the following:

This means 248968 ≈ 1.475830912 × 1035

Summary of the Graph of Exponential Functions in the form 𝒇(𝒙) = 𝒃𝒙𝒐𝒓 𝒚 = 𝒃𝒙

The domain of 𝒇(𝒙) = 𝒃𝒙is all real numbers

The range of 𝒇(𝒙) = 𝒃𝒙 is all positive real numbers, 𝒇(𝒙) > 𝟎 𝒐𝒓 𝒚 > 𝟎

The graph of 𝒇(𝒙) = 𝒃𝒙 must pass through the point (0,1) because any number, except zero, raised to the

zero power is 1. The y-intercept of the graph 𝒇(𝒙) = 𝒃𝒙 is always 1.

The graph of 𝒇(𝒙) = 𝒃𝒙 always has a horizontal asymptote at the x-axis (𝒚 = 𝟎) because the graph will get

closer and closer to the x-axis but never touch the x-axis.

If 𝟎 < 𝒃 < 𝟏 the graph of 𝒇(𝒙) = 𝒃𝒙 will decrease from left to right and is called exponential decay.

If 𝒃 > 𝟏 the graph of 𝒇(𝒙) = 𝒃𝒙 will increase from left to right and is called exponential growth.

The value of the growth factor, 𝒃, determines whether an explicit formula is modeling exponential growth or

exponential decay.

If 𝒃 > 𝟏, output will grow over time. If 𝟎 < 𝒃 < 𝟏, output will decay over time.

If 𝒃 = 𝟏 the output would neither grow nor decay; the initial value would never change.

Exponential Growth

The exponential function with base 𝒃, also known as the growth factor, is defined by

𝒚 = 𝒂 ∙ 𝒃𝒙 where 𝒃 > 𝟏, 𝒃 ≠ 𝟏, and 𝒙 is any real number. The initial amount, when 𝒙 = 𝟎, is represented by 𝒂.

*When dealing with percentages, 𝒃 is equal to 1 plus the percent rate of change expressed as a decimal.

Exponential Decay

The exponential function with base 𝒃, also known as the decay factor, is defined by

𝒚 = 𝒂 ∙ 𝒃𝒙 where 𝟎 < 𝒃 < 𝟏, 𝒃 ≠ 𝟏, and 𝒙 is any real number. The initial amount, when 𝒙 = 𝟎, is represented by 𝒂. *When dealing with percentages, 𝒃 is equal to 1 minus the percent rate of change expressed as a decimal.

The domain is the set of all first elements of ordered pairs (x-coordinates).

The range is the set of all second elements of ordered pairs (y-coordinates).

Functions can have "hills and valleys": places where they reach a

minimum or maximum value.

It may not be the minimum or maximum for the whole function, but

locally it is.

A Vertical Line Test is used to determine if a relation is a function. A

relation is a function if there are no vertical lines that intersect the

graph at more than one point.

Name Definition Formulas

Simple Interest

Interest is calculated once per year on the original amount borrowed or invested. The interest does not become part of the amount borrowed or owed (principal).

𝐼 = 𝑃𝑟𝑡 𝐼 = the interest earned after t years 𝑃 = the principal amount (amount borrowed or invested) 𝑟 = interest rate in decimal form

Compound Interest

Interest is calculated once per period on the current amount borrowed or invested. Each period, the interest becomes part of the principal.

𝐴 = 𝑃 (1 +𝑟

𝑛)

𝑛𝑡

𝐴 = the future value 𝑃 = the present value 𝑟 = interest rate as a decimal 𝑛 = the number of times compounded per year 𝑡 = time in years

Linear Model Exponential Model

General Form 𝒇(𝒙) = 𝒂𝒙 + 𝒃 𝒇(𝒙) = 𝒂 ∙ 𝒃𝒙

Meaning of

parameters 𝒂

and 𝒃

𝒂= rate of change

𝒃=initial value (when 𝒙 = 𝟎)

𝒂=initial value (when 𝒙 = 𝟎)

𝒃= rate of change

*Note: Intervals represent the 𝒙-values (or the input values)

Name Definition Representation

Function A function is a correspondence between two sets, 𝑋 and 𝑌, in which each element of 𝑋 is matched to one and only one element of 𝑌. The input values of a function must UNIQUELY map to one output value

*5 maps to more than one output – not a function

Domain The domain refers to the “X” values of the function (the input)

For the function above: Domain = {2, 3, 4, 5}

Range The range refers to the “Y” values of the function (the output)

For the function above: Range = {4, 5, 8}

Function Notation

Traditionally, functions are referred to by the letter name f, but they can be referred to by other letters. The f (x) notation can be thought of as another way of representing the y-value in a function, especially when graphing. The y-axis is even labeled as the f (x) axis, when graphing.

Name Definition Representation

Piecewise – Linear Function

Given a number of non-overlapping intervals on the real number line, it is the union of the intervals to the set of real numbers such that the function is defined by linear functions on each interval.

Absolute Value Functions

The absolute value of a number x, denoted by |x|, is the distance between 0 and x on the number line. It is a piecewise function such that for each real number x the value of the function is |x|.

Step Functions A step function is a special type of function whose graph is a series of line segments. The graph looks like a series of small steps.

Inside: Horizontal Outside: Vertical

𝑓(𝑥 + 1) Left 1 𝑓(𝑥) + 1 Up 1

𝑓(𝑥 − 1) Right 1 𝑓(𝑥) − 1 Down 1

𝑓(2𝑥) Horizontal shrink (gets narrower

– scale factor 1

2)

2𝑓(𝑥) Vertical stretch (gets narrower –

scale factor 2)

𝑓 (1

2𝑥)

Horizontal stretch (gets wider –

scale factor 2)

1

2𝑓(𝑥)

Vertical shrink (gets wider –

scale factor 1

2)

Name Definition Examples

Sequence An ordered list of elements that change

according to some sort of pattern. 1, 3, 5, 7, 9, …

Terms of the Sequence The elements of the list. 1st term: 1 3rd term: 5 2nd term: 3 4th term: 7

Indexed The terms are ordered by a subscript

starting at 1.

𝑎1 = 1, 𝑎2 = 3, 𝑎3 = 5, 𝑎4 = 7, … “…” means that the pattern is regular

and continues.

Name Definition Examples Things to Know

Explicit Formula

Specifies the nth term of a sequence as an expression in n. You need to know what integer you are using for the first

term number to write an explicit formula.

f(n) = 12 – 3(n – 1) starting with n = 1

Can find any term, any time

Recursive Formula

Specifies the nth term of a sequence as an expression in the previous term (or previous couple terms). It is a

sequence that (1) is defined by specifying the values of one or more initial terms and (2) has the property that the remaining terms satisfy a recursive formula that describes the value of a term based upon an expression in numbers,

previous terms, or the index of term. You need to know what the first term is, or first several terms are, depending in the recursive relation to write a

recursive formula.

51 nn aa , where

121 a and 1n

Uses previous term to find the next

term

Name Definition Examples Formulas

Arithmetic Sequence

A sequence is called arithmetic if there is a real number d such that each term in the sequence is the sum of the previous term and d. These are often referred to as a “linear sequence”.

14, 11, 8, 5, … (Minus 3 each time)

Geometric Sequence

A sequence is called geometric if there is a real number r such that each term in the sequence is a product of the previous term and r.

2, 10, 50, 250, … (Multiply by 5 each time)