Algebra 2 Semester 1

Inconsistent system

No solutions

If you know the slope & one of the coordinates on a line, you can use ... to find the equation of the line

Point Slope Form

if |a|=b then...

a = b or a = -b *av must always be isolated *ALWAYS CHECK BACK INTO EQUATION for extraneous situations

Line of fit

a line that closely represents a set of data

Rate of Change

a ratio that compares how much one quantity changes relative to the change in another quantity - slope

inverse property

a+(-a) = 0 .......... a·1/a = 1 (a can't be 0 for multiplication)

identity property

a+0 = a .......... a·1 = a

commutative property

a+b = b+a .......... a·b = b·a

closure property

a+b = real number a·b = real number

reflexive property

a=a

regression line

absolute best equation to represent scatter plot (on graphing calculator)

Prediction equation

an equation for the line of fit

rational numbers

any number that can be written as a/b where a & b are both integers and b is not 0. The decimal form will repeat or terminate.

distributive property

a·(b+c) = ab + ac

vertex points

corners of feasible region that show potential points of optimization

irrational numbers

decimals that go on forever but can't be written in a/b form when a & b are integers

Absolute value

distance from 0 always will be positive treat like (...) - do first

addition and subtraction property

do same to both sides when adding/subtracting

multiplication and division property

do same to both sides when multiplying/ dividing

X value terms

domain, input values, independent variables

Bounded Feasible Region

enclosed graph- max and min value of a function will ALWAYS occur at the vertex of the feasible region

Parallel Lines Slope Relation

exact same slope

Function Notation

f(x) - implies we want to take this function and evaluate it for any possible x value

FUNCTION multiplied by negative number (usually -1)

flip over x axis `

X multiplied by negative number (usually -1)

flip over y axis

One to One Function

for every range, you can only have one domain (Y values don't repeat)

>

greater than

What to do if graphing linear programming and can't pick out a distinct vertex

set them up as system of equality

Feasible Region

shaded region in linear programming

"OR" compound inequalities

solution set is combination of the two solutions- represents any possible number that satisfies EITHER answer

"AND" compound inequalities

solution set is overlap of the two solutions - represents possible answers that satisfy BOTH answers

Function

specific domain, range relationship where every domain has only one range value (X values don't repeat)

when r is close to -1

strong negative correlation

when r is close to 1

strong positive correlation

algebraic expression

sums, differences, products and/or quotients of numbers and variables (NO EQUAL SIGN)

How to find vertex for absolute value function

take what's inside of the absolute value bars and set it equal to zero - plug the x value you get back into your equation to find the y value

parent function

the base graph for a particular type of equation

Slope Intercept Form

this is the only way an equation can show the true slope of a line

Compound inequality

two inequalities joined together by "and" or "or"

FUNCTION multiplied by number between -1 and 1

vertical compression

positive number subtracted from FUNCTION

vertical slide down

positive number added to FUNCTION

vertical slide up

FUNCTION multiplied by number bigger than 1 or less than -1

vertical stretch

verbal expression

writing out algebraic expressions with words in sentence format

"Less than" Absolute value inequality * |x|<a

x<a AND x>-a

"Greater than" Absolute value inequality * |x|>a

x>a OR x<-a

constant function

y= a number

Quadratic function

y= x^2

Linear/ identity function

y=x (slope intercept)

Absolute value function

y=|x|

interval notation

( means not included ) [ means included ]

Graphing inequalities rules

< or > : dashed line ≤ or ≥ : solid line

Consistent system

At least one solution

Standard Form

Ax+By=C - A,B, and C must be integers with a GCF of 1 and A≥0

Vertical Line test

If a vertical line intersects a graph at more than one point then it's not a function

symmetric property

If c= a+b then a+b= c

steps to graphing linear/ absolute value inequalities

1. get y alone on one side- simplified 2. graph the line* 3. Pick a point and plug into inequality 4. If it's true then shade that side of the line, if not shade the other side *if an absolute value it will be shaped like a V

how to solve system of inequalities

1. graph and shade first inequality 2. graph other inequalities 3. identify region of graph that's double shaded

How to solve by substitution

1. pick an equation and isolate the variable 2. substitute the resulting equation in for that variable in the other equation 3. solve for the variables

associative property

(a+b)+c = a+(b+c) .......... (a·b)·c= a·(b·c)

How to find line of best fit w/ graphing calc.

*1. STAT → ENTER (put data in chart) 2. adjust WINDOW 3. set up scatter plot 2nd → Y= → ENTER 4. GRAPH *5. for info on reg. line click STAT → "right arrow" → 4 → ENTER 6. to see line, click Y= and enter equation

Solving inequalities

*If divide or multiply by negative then flip the inequality symbol *graphed on number line - open circle if just greater or less than - filled in circle if greater than or equal to OR less than or equal to *infinite # of solutions possible

How to set up real world linear programming scenarios

- Variables - Constraints - Function - Vertices - Optimized scenario

How to solve by elimination

- align the coefficients of one of the variables in both equations - if coefficients are the same : subtract the equations - if coefficients are opposite values (negative and positive): add the equations

Unique Scenarios for Substitution and Elimination way of solving inequalities

- if variables cancel out and you're left with a false statement then no solution exists - if variables cancel out and you're left with a true statement then there are infinite solutions

Step/ Greatest Integer Function

- written as [[x]] - many small segments that look like stairs - right side of segment is open circle, left is closed - start with x=0 and slowly work away from y axis - Domain will be: (-∞,∞) - Range will be: All integers

3 methods for solving systems of equations

-Graphing -Substitution -Elimination

Piecewise functions

-individual rays and segments that make a graph when put together (uses inequalities) - use an open circle to show that a point is not included - use a closed circle if a point is included - work from L to R

Constraints

-represented by inequalities - boarders of shaded region when graphed

How to find line of best fit w/o graphing calc.

1. Select 2 points that represent the data well 2. draw the line on the graph 3. use the two points to find the slope 4. put info in point-slope form 5. simplify to slope intercept form

how to solve triple variable systems using elimination

1. find easy looking variable 2. take the first two equations ans eliminate that variable then take the second two equations and eliminate the same variable 3. take the results from step two and eliminate one of the two remaining variables 4. solve for the equations that's left and plug it in to find the other two variables

system of equations

A scenario where 2 (or more) equations consist of the same variables

Independent system

A system with exactly one solution

Dependent system

A system with infinite solutions

Y value terms

range, output values, dependent variables

Perpendicular Lines Slope Relation

The slopes must be opposite reciprocals of each other / product of the slopes is -1 - example: line perpendicular to 3 would be -1/3

X multiplied by number bigger than 1 or less than -1

horizontal compression

positive number added to X

horizontal slide left

positive number subtracted from X

horizontal slide right

X multiplied by number between -1 and 1

horizontal stretch

substitution property

if 4(a)=b and a=7 then 4(7)= b

Horizontal line test

if a horizontal line intersects a graph at more than one point then it's not a one to one function

Finding vertices in system of inequalities

if enclosed region - determine coordinates where boundary lines intersect

Discrete relations

individual points on graph that are not connected on a graph

Continuous relations

infinite number of points that will be graphed as a smooth continuous curve

system of inequalities

infinite possible combination of variable values

<

less than

Linear Relation (4 rules)

line as a graph - no variables multiplies - no variable in any denominator - no exponents (other than 1) on any variable - no variables inside of any roots

when r is close to 0

no correlation

integers

no decimals

whole numbers

no negatives, no decimals

natural numbers

no negatives, skip zero, no decimals

Unbounded Feasible Region

open graph that goes on forever- contains max or min but NEVER both

Unique scenarios for graphing inequalities

parallel lines - no solution exact same line - infinite solutions

Unique scenario when graphing systems of inequalities

parallel lines shaded away from each other- no solution

Linear Programming

process of optimizing scenarios

Correlation Coefficient

r-value on graphing calculator- measures how well scatter plot is modeled by a linear equation