Chapter 2 Algebra 2

Vertical Lines

-Written in the form x=a, where a represents the line's x-intercept -Undefined slope

Horizontal Lines

-Written in the form y=a, where a represents the lines y-intercept -Zero slope

Steps to a solution

1. Graph the system of contraints (inequalities) 2. Name corner points on a graph 3. Use a system of equations to accurately find exact coordinates of corner points 4. Complete max/min corner point chart 5. Identify the max or min 6. Report Out

Writing equations perpendicular to a given equation that passes through a point

Use the slope from the given equation and FLIP and NEGATE it to make it perpendicular. Then use that slope and the given point and put it into y-y1=m(x-x1)

Writing equations parallel to a given equation that passes through a point

Use the slope from the given equation and the given point and put it into y-y1=m(x-x1)

Given a point and the slope, how do you put it in slope-intercept form?

Use y-y1=m(x-x1) and put in the points for y1 and x1 and put in the slope for m. Then simplify.

% word problem chart systems

amount % total 1 2 mix add amount 1+amount 2= mix amount multiply across to get totals, and then do total1+ total2= mix total

c is _____ c= _____

constant, all real numbers

Graphing INEQUALITIES

graph as usual, plug in 0 in each equation to determine where to shade. If 0 works, shade towards it, and then determine the overlap

f(cx) where c>1

horizontal shrink by a factor 1/c how: multiply x-values by 1/c

f(cx) where 0<c<1

horizontal stretch by a factor 1/c how: multiply x-values by 1/c

Vertical LIne Test

if a vertical line passes through more than one point on the graph of a relation, then the relation is not a function

Vertex principle of Linear Programming

if there is a minimum or maximum of the linear objective function then it occurs at one of the corner points of the feasibility region

Function Notation

input = x output = f(x) f(x)=3x+1 input=output

Solving Standard Form

method 1: Solve for y plot using slope intercept method 2: letx=o; find y intercept let y=o; find x intercept Plot points connect ARROWS

Graph transformations: apply to all ____ graphs inside the absolute value= ____ outside the absolute value= ____

parent horizontal change vertical change

-f(x)

reflection about x-axis how: multiply y-values by -1

f(-x)

reflection about y-axis how: multiply x-values by -1

Evaluating functions

replace x with anything that is in the parenthesis

f(x)-c

shift down c units how: subtract c from y-values

f(x+c)

shift left c units how: subtract c from x-values

f(x-c)

shift right c units how: add c to x-values

f(x)+c

shift up c units how: add c to y-values

m=

slope

objective function

the expression/equation we are trying to minimize or maximize

Constraints/Boundary Lines

the inequalities that create the shaded region

Domain

the possible input values (x-values) of a relation or function

Range

the possible output values (y-values) of a relation or function

feasibility region

the shaded region

Ordered Pairs

(1.2) (3,4) (3,5)

Using Calculator steps to plot a table/line of best fit

1. STAT 2. Edit 3. Write values for x in column 1 and y in column 2 4. 2ND, Y=, Make sure STAT plot is on 6. WINDOW, make x and y min and max big enough 7. hit GRAPH Line of best fit 1. STAT 2. CALC 3. LinReg 4. Enter, enter again 5. Look at A for slope, B for Y intercept to get the equation 6. then if r is close to -1 or 1, the line of best fit is a good fit Make sure Diagnostic is on by pressing 2ND, 0, turn it on, enter

Solving linear systems using substitution

1. choose an equation and solve for one variable 2. substitute into the other equation 3. solve 4. use your solution in the other equation to find the other variable 5. check your answer (if never true in end, no solution)

order of describing transformations

1. left or right 2. horizontal stretch, shrink, or reflection (y-axis) 3. vertical stretch, shrink, or reflection (x-axis) 4. up or down

solving linear systems using elimination

1. stack the equations and get the like terms on top of each other 2. multiply the entire equation by a constant to get one variable to have opposite coefficients if needed 3. add the equations together to eliminate a variable 4. substitute the value of the variable into either original equation to solve for the other variable 5. check answer if 0=0 or always true, infinite solutions

how to graph absolute value equations

1. start with parents table (-2, 2) (-1, 1) (0,0) (1,1) (2,2) 2. Determine the sequence of transformations, and use each ones "hows" to edit table until you have done all transformations 3. Graph final graph 4. Write Domain and Range (D= left and right, R= lowest and highest. infinity is always (, a vertex is always a ] )

Functions

A relation in which each x-value corresponds with exactly one y-value

Standard form

Ax+By=C

Domain and range Interval Notation

Ex. x= 1,6,1 y= 2,5,6 D: { 1,6} R: { 2,5,6} Practice on Graphs too!!!

Given two points, how do you put it in slope-intercept form?

Find the slope using y2-y1/x2-x1. Then pick one of the points you were given and plug them and the slope into y-y1=m(x-x1)

Parallel Lines

Parallel lines have the same slope different y intercept

Perpendicular Lines

Perpendicular lines have negative reciprocal slopes

Solving Linear Equation Systems by Graphing one solution no solution many solutions

Put all into y=mx+b, graph, and any ordered pair that makes the equations true within the system is a solution. Where they intersect is the answer one- when lines intersect no- lines are parallel Many- Lines are on top of each other (infinite solutions)

how to describe the transformations made to get from one equation to the other

use math to see where you may need to add or subtract numbers to get from one number to another. the change ( # ) and its location will determine that move

cf(x) where 0<c<1

vertical shrink by a factor of c how: multiply y-values by c

cf(x) where c>1

vertical stretch by a factor of c how: multiply y-values by c

corner points

where the boundary lines intersect at the edges of the feasibility region

With wind/current vs against

with (x+y) against (x-y) time * rate= distance

b=

y intercept

Point-Slope Formula

y-y1=m(x-x1)

How to find Slope given two points

y2-y1/x2-x1

Parent equation

y= I x I I I is a grouping symbol, treated like parenthesis

Direct Variation

y=kx or y/x=k the ratio of all output-input pairs equals the constant k we call the constant of variation ex. x y 1 2 3 6 4 8 2/1=6/3=8/4 =2 so y varies directly w/ x and k=2 so y=2x

Slope-Intercept Form

y=mx+b