Algebra 2 Unit 1 Linear Equations & Systems

Variable

Symbol, ie. letter, represents 1+ number(s)

Ex: n

Ex: x

Numerical expression

Math phrase, numbers and operation symbols

Ex: 3+5

Ex: (8-2)+5

Algebraic Expression

math phrase, 1+ variables

Ex:

3n+5

(8x-2)+5n

Always true solution

Ex: 2=2

Sometimes true solution

x=2

Never true solution

2=3

|x|

Distance from 0

\+

Absolute value

|x| verbal

distance of # from 0

|x| graph

y=|x|

|x| table

|X| Analytical

{-x x<0 m=-1

f(x)= perpendicular

{x x≥0 m=1

\

Extraneous solutions

1+ answer

check if both actually correct

both, 1, or neither

when Xs both in and outside absolute value

substitute into og equation

Interval Notation

\#<x<#

\#>x>#

Infinite #s between, irrational, fractions

(,)

Ex:

\-7<x<10

(-7,10)

Interval notation

\#≥x≥#

\#≤x≤#

\[,\]

ex:

3<x≤205

\[3,205\]

Interval notation infinity

Ex:

x>-1

(-1, ∞)

Infinity always parenthesis, can’t stop at ∞

interval notation U

Union, “or”

interval notation graph

\

|x|> -#

|x| always > negative #

And statement x

\#<x<#

And statement

greater absolute value inequality

GREAT OR

Ex:

|5x+10|>15

5x+10>15 or 5x+10<-15

Absolute value parent functions table

Absolute value parent functions function

{|x|=x, when x≥0

f(x)=

{|x|=-x, when x<0

Absolute value parent functions graph

Absolute value parent functions vertical translation

up k units, k>0

y=|x|+k

\
down k units, k>0

y=|x|-k

Absolute value parent functions horizontal translation

right h units, h>0

y=|x-h|

\
left h units, h>0

y= |x+h|

Absolute value parent functions vertical stretch & compression

stretch, a>1

y=a|x|

\
compression, 0<a<1

y=a|x|

Absolute value parent functions reflection

x-axis

y=-|x|

\
y-axis

y=|-x|

Parent function

y=|X|

General form of absolute value function

y=a|x-h|+k

\

a

y=a|x-h|+k

stretch/compression factor

|a|

vertex

y=a|x-h|+k

(h,k)

axis of symmetry

y=a|x-h|+k

x=h line

relation ordered pair

(input, output)

(x,y)

Ex:

(-3,4)

(3,-1)

(4,-1)

(4,3)

relation mapping diagram

\

relation table of values

relation graph

Function notation

f(x)=y=3x^2+√x+34x^3

x:input, independent

y:output, dependent

\
Ex: f(x)=y

f(x)=3x^2-x+1

f(-1)=3(-1)^2-(-1)+1

f(-1)=3+1+1=5

Function rule

equation that represents output in terms of input values

Can write in function notation

Ex:

y=3x+2

f(x)=3x+2

f(1)=3(1)+2

Domain

set of all inputs (x values)

range

set of all outputs (y values)

Domain and range Ex

f(x)=y=3x-4

f(pi)=3pi-4

\
D: (-∞, ∞)

R: (-∞, ∞)

function

special relation

each element in domain(x) corresponds to EXACTLY 1 value in range(y)

Each input 1 output

vertical line test

vertical line crosses graph 1+, not function

Maximum function

largest Y VALUE

Minimum function

smallest Y VALUE

zeros functions

x-intercepts (Cross x axis)

linear parent function

f(x)=x

quadratic parent function

f(x)=x^2

cubic parent function

f(x)=x^3

absolute parent function

f(x)=|x|

reciprocal parent function

f(x)=1/x

Exponential parent function

f(x)=e^x

logarithmic parent function

f(x)=ln x

square root parent function

f(x)=√x

Solve ≥ equation

1. simplify PEMDAS
2. Solve inequality PEMDAS

Ex:

\-3(2x-5)+1≥4

\-6x+15+1≥4

\-6x+16≥4

\-6x≥-12

x≤2

Solve & Equation

Solve individually

Ex:

7<2x+1 & 3x≤18

6<2x & x≤6

3<x & x≤6

3<x≤6

Solve or equation

Solve individually

Ex:

7+k≥6 or 8+k<3

k≥-1 or k<-5

Solve absolute equality

1. simplify pemdas
2. 2 equations one with a negative side without x
3. solve individually

Ex:

3|x+2|-1=8

3|x+2|=9

|x+2|=3

x+2=-3 or x+2=3

x=-5 or x=1

Interval notation

\#<x<#

\#>x>#

(#,#)

least to greatest

Ex:

3<x≤205

(3,205\]

x between 2 points

Interval notation

X>#

X<#

(-∞,#)

(#,∞)

Infinite side always (), can’t stop at infinity

x extends in one way forever

Interval Notation

()

\[\]

>/<

≥/≤

Flip inequality sign

Divide by -#

Solve Equality

1. Simplify
2. Split into 2 equations if there is a negative
3. Solve by PEMDAS

y≤ line graph

solid line

shade below

y< line graph

dotted line

shade below

y≥ line graph

solid line

shade above

y> line graph

dotted line

shade above

Solution to inequality system graph

Overlapping region

Solve 3 system

1. Simplify
2. Find one variable
3. Substitute


1. Solve by PEMDAS

For any value of x

you can plug in any value for x and it would stay the same

Slope

Nonvertical line through (x1,y1) and (x2,y2)

Ratio of vertical to horizontal change

rise/run

y2-y1/x2-x1, where x2-x1=/=0

m

horizontal slope

y=#

slope=0

vertical slope

x=#

slope undefined

\+ Slope

increasing, rising

\- Slope

decreasing, falling

Slope-intercept form

y=mx+b

\
m:slope

b: y-intercept

Point-slope form

y-y1=m(x-x1)

\
through point (x1,y1)

m:slope

\
substitute (x,y) for (x2,y2), rewrite slope formula

Standard form

Ax+By=C

\
A, B, and C: integer constants

A: >0

Scatter plot

graph

relates 2 data sets

plot data as ordered pairs

determine strength of relation/correlation between sets

Determine relationship strength scatter plot

stronger relations form line, stronger +/- correlation between values

System

multiple equations

Solution to system

points (if any) in common

point where they meet on a graph

Different slopes (Independent) system

One solution

Parallel (Inconsistent) system

Parallel (no solution)

Same slope

Diff. y-intercept

Same line (Dependent) system

Same line (infinitely many solutions)

Same slope

Same y-intercept

Graphing solve system

Easy but slow

Problematic

Substitution solve system

1. Solve for 1 variable in 1 equation
2. Plug expression into other equation

Elimination solve system

add/subtract equations and keep equality

3-variable equation

intersection of 3 planes

Natural number

positive whole number

Integer and whole

Integer

Positive and negative #≥1 and 0

rational

Whole number

Positive #≥1 and 0

integers

Rational number

can be written as a fraction

Irrational number

cannot be written as a fraction

Real number

can be used to measure quantity

can be put on a number line

Properties of equality

1. reflexive: a=a
2. symmetric: a=b, b=a
3. transitive: a=b, b=c, then a=c
4. addition: a=b, then a+c=b+c
5. subtraction: a=b, then a-c=b-c
6. multiplication: a=b then a*c=b*c
7. division: a=b, then a/c=b/c
8. substitution: a=b, then b can be substituted for a

Properties of multiplication

1. Commutative: a*b=b*a
2. associative: a(bc)=(ab)c
3. identity: a*1=a, b*1=b, etc.
4. zero property: a*0=0, b*0=0, etc.
5. distributive: a(b+c)= ab+ac

Properties of division

1. 1: a/1=a, b/1=b, etc.
2. itself: a/a=1, b/b=1, etc.
3. # by 0: a/0=undefined, b/0=undefined
4. 0 by #:0/a=0, 0/b=0, etc.

Properties of subtraction

1. identity: a-0=a
2. Commutative: a-b=/=b-a
3. itself: a-a=0

Properties of addition

1. associative: (a+b)+c=a+(b+c)
2. commutative: a+b=b+a
3. identity: a+b=b+a, a+0=a