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
