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) = 5 3 . 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 0 =1 Product Property of Exponents ∙ = + Power of a Quotient Property ( ) = Definition of Negative Exponents − = 1 or ( ) − =( ) Power of a Power Property ( ) = Power of a Product Property () = Common Base Property of Equality If = , and ≠ 1, then = . Quotient Property of Exponents = − Power Property of Equality If = , then = . Equivalence of Radicals and Rational Exponents 1 = √ and = √ = ( √ ) 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 × 10 4 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 =4, 1 = 616 This means 6 × 10 16 **THESE ARE ROUNDED VALUES** When we try to calculate 24896 8 , we get the following: This means 24896 8 ≈ 1.475830912 × 10 35
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
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.