Algebra 2 Sol Things To 3s5233

Algebra 2

SOL Things to  Test-taking strategy

Graphing Calculator Tips – negative numbers in parenthesis; fractions in parenthesis; numerator in parenthesis and denominator in parenthesis Properties Properties

In words

Algebra

Commutative*

Switch order

a + b + c = a + c + b ; a×b×c = a×c ×b

Associative*

Switch grouping

( a + b ) + c = a + ( b + c ) ; ( ab ) c = a ( bc )

Identity

Back where we started from

a + 0 = a; a ( 1) = a

Inverse

Back to the identity

 1 a + ( −a ) = 0; a   = 1 a

Distributive

Addition over multiplication

a ( b + c ) = ab + ac

Reflexive

a=a

Symmetric

a = b, then b = a

Transitive

a = b, b = c, then a = c

*Commutative and Associative of Multiplication doesn’t apply to Matrices. Solving Equations Absolute value

ABS - under MATH and NUM

> or ≥ - shaded “out” > a negative number = all real numbers < a negative number = no solution

< or ≤ - shaded “in”

Other equations  

Put one side in Y1 and the other side in Y2 and find the intersection. Plug answers in to find the one that satisfies the equation.

Factoring a2 − b 2 = ( a + b ) ( a − b ) Difference of Perfect Squares

( = (a

)( ) ) ( a + b) ( a − b)

a4 − b4 = a2 + b 2 a 2 − b 2

Sum and/or Difference of Perfect Cubes

2

+b

2

(

a 3 ± b 3 = ( a ± b ) a 2 mab + b 2

)

a = root; ( x − a ) = factor; where it crosses the x-axis Double root – function just touches the x-axis



Only one variable? Plug in the original into Y1 and plug the answers into Y2. Graph and find the one that matches the original.  Plug in values for the variables. Avoid plugging in 0 and 1.

Functions Types of Functions Function

Sample Equation

Sample Graph

Regression

Linear

y = 2x + 5

LinReg

Quadratic

y = x 2 − 3x + 4

QuadReg

Cubic

y = 3x 3 − 2x + 1

CubicReg

Quartic

y = 2x 4 − 3x 2 − x + 1

QuartReg

Absolute Value

y = 3x − 4

Step

y =  x 

Exponential

y = 2x

ExpReg

Logarithmic

y = log2 x

LnReg

Evaluating - f (a ) ; plug in a into the function for x Plug function into y1 and look at the table Composition - f ( g ( x ) ) or

[ f og ] ( x )

Plug in g ( x ) for x in the function f ( x )

Inverse – switch x and y and solve for y; opposite operations; reflected over the line y = x Translations f (x ± c )

Left (+) c units; Right (-) c units

f (x ) ± c

Up (+) c units; Down (-) c units

af ( x )

Multiply all y-values by a

f (ax )

Divide all x-values by a

−f ( x )

Flips over the x-axis; multiplies all y-values by -1

f (− x )

Flips over the y-axis; multiplies all x-values by -1

Domain and Range Domain = input; no zeros in the denominator; no negative numbers under the radical (even roots) Range = output; watch out for fractions; exponentials; radicals  Plug into the calculator and use the TABLE feature.  Look at the graph – “squash” to the x to find domain; “squash” to the y to find the range. Exponent Rules Multiplying with like bases – add the exponents

(x )(x ) = x

Dividing with like bases – subtract the exponents

xa = x a−b xb

a

b

Raising to a power – multiply the exponents

(x )

b

a b  

−1

Raising to -1 – flip Fractional exponents*

xb =

a

a+ b

= x ab b a

or

xa =

( x)

=

a

b

b

a −1 1 = b ab a

*There have consistently been questions on Fractional exponents on each Algebra 2 SOL.

i i2 i3 i4



i -1 −i 1

Powers of i Conjugate – multiply by the conjugate to eliminate radicals from the denominator conjugate = 2 − 7 2+ 7 3 − 2i conjugate = 3 + 2i

the i key; Reset the calculators MODE to a + bi

Systems of Equations Matrix multiplication – can only multiply if the number of columns in the first = number or rows in the second 2 x 4 by 4 x 5 - result is 2 x 5

[ A]

−1

- find the inverse of matrix A

Ax + By = C Dx + Ey = F

Matrix Equation

 A B   x  C  D E   y  =  F      

−1

 x   A B  C   y  = D E  F       

Plug the answers into the equations to see if they satisfy both. Solve for y and graph both. Solution is where the equations intersect. Use the PlySmlt2 APP

  

Systems of Inequalities  Solve for y and graph in the calculator. to flip the sign when multiplying or dividing by a negative.  Use Inequality APP  Check the lines - < or > should be dashed; ≥ or ≤ should be solid  When solved for y: shaded above - > or ≥ ; shaded below - < or ≤ Linear Programming Find the corner points and PLUG into the given equation. Don’t guess, actually plug them in!

 Variation Direct

Divide

y varies directly x

y =k x

Inverse

Multiply

z varies inversely with x

zx = k

t

Divide and multiply

s varies directly with x and inversely with m

sm =k x

Sequences and Series Arithmetic = add the same thing each time Geometric = multiply by the same thing each time

∑

end

∑ rule

= find the sum; write them out and add them up!

start



Plug the formula into Y1 and go to the table to find the values to add up

Statistics Scatter plots -

STAT

EDIT

STAT Plot (setup/turn on)

ZOOM

9 (STAT)

Decide what type of function (see the types above) it is in order to choose the appropriate equation to find. STAT CALC Choose the function to match your graph. Don’t guess! Use your calculator!! How to calculate the percentage under the normal curve

1.

Pressing 2nd, VARS, and 2:normalcdf( .

Enter the lower end of your range, the upper end of your range, the mean ( µ ) and the standard deviation ( σ ).

2. 3.

Press ENTER.

Permutations – order makes the set different; your locker combination is a PERMUTATION

total objects MATH PRB 2: nPr number of objects in subset Combinations – order doesn’t matter; same set even in a different order; ex. toppings on a pizza total objects MATH PRB 3: nCr number of objects in subset Conic Sections Identify

Equation

Sample Graph

y = a ( x − h) + k 2

Parabolas

Ellipse

One squared term

Two squared ; addition; different coefficients

x = a( y − k) + h 2

( x − h)

2

+

a2

( x − h)

2

+

b2

Circle

Two squared ; addition; same coefficients

( x − h)

2

( x − h) Hyperbola

Two squared ; subtraction

( y − k) a2

2

=1

b2

( y −k)

2

=1

a2

+( y − k) = r2 2

2

−

a2

( y −k)

2

−

( y − k)

2

=1

b2

( x − h) b2

2

=1