Math B Study Guide-Under Construction

Circles  Complex Numbers   Coordinate Geometry Proofs    Exponential    Fractions       Functions  Geometry Proofs

Logarithms          Probability        Quadratic Formula      Radicals  Regressions         

      Statistics       Transformations   Trigonometry                    

Solving Absolute Value Equations/Inequalities       Solving Radical Equations         Solving Quadratics Equations

 

 

Quadratic Formula

For ax2 + bx + c = 0 where a, b, and c are real numbers, and a Ή 0

            Axis of symmetry

Sum of Roots Product of roots

Discriminants

Value of the Discriminant

Nature of the Roots

1) b2 – 4ac > 0 and a perfect square

1) roots are real, rational, and unequal

2) b2 – 4ac > 0 and not a perfect square

2) roots are real, irrational, and unequal

3) b2 – 4ac = 0

3) roots are real, rational, and equal

4) b2 – 4ac < 0

4) roots are imaginary

 

 

Logarithms & Exponents

Logarithm as an exponential expression: log b x = y « by = x

 

Special Rules:

log b 1 = 0 « because b0 = 1

log b b = 1 « because b1 = b

log b ba = a « because ba = ba

Computations with Logarithms

Product Rule: log b AB = log b A + log b B

Quotient Rule: log b = log b A - log b B

Power Rule: log b Ac =c(log b A)

Solving a Logarithm Equation:

1st Way log b (x +2) = log b x2 (same base on both sides)

1) Write an equation by setting equal the 2 values your finding the log of.

2) Solve the resulting equation and check

2nd Way Log27=x

1) Rewrite the logarithm as an exponential equation

2) Solve as you would an exponential equation

Solving Exponential Equations:

1st Type: 42x = 8x+2

1) Get like bases

2) Multiply exponents if necessary

3) Set Exponents equal to each other

4) Solve the equation and check

2nd Type:

1) Raise both sides to an exponent that will make the given exponent of the x equal to 1

2) Solve equation and check

3rd Type: 4x + 1 = 7 (can’t get same base)

1) Do the logarithm of both sides

2) Use the power rule for logarithms, bring the exponent down front.

3) Solve the equation for the variable and check

Graphing a Logarithmic or Exponential Function:

1) Either use the interval given or set up own interval by using negatives, zero, and positive numbers

2) Plot points and sketch smooth curve.

3) Label graph(s)

Note: The inverse function of a logarithmic equation is an exponential equation, the reflection over the line y = x

ex. y = log b x inverse to y = bx

 

Functions

Relation is a Function:

Passes the vertical line test     or

For any x value in the domain there is exactly one y-value in the range

One to One Function:

Passes the horizontal line test     or                                

For any x value in the domain there is exactly one y-value in the range and for any y-value in  the range there is exactly one x-value in the domain.

Inverse of a Function:

1) Exchange you x and y.

2) Solve new equation for y.

Graph of an Inverse Function: Reflect over the line y = x

Graphs of Equations your should know:

Linear Function

y = mx + b is the equation of a line that is a function. Ex. x + 2y = 3

Quadratic Function

y = ax2 + bx + c is the equation of a parabola. Ex. y = x2 – 4x

Conics that are Relations

**x = ay2 + by + c is the equation of a parabola Ex. x = y2 –4y

ax2 +by2 = c if a = b and a, b, c have the same signs is the equation of a circle. Ex. 2x2 + 2y2 = 8

 

ax2 +by2 = c if a Ή b and a, b, c have the same signs is the equation of an ellipse. Ex. 2x2 + 3y2 = 6

ax2 +by2 = c if a Ή b and a, b, c have different signs is the equation of a hyperbola. Ex. 9x2 – 3y2 = 9

Hyperbola that is a function xy=c also called an inverse relation

Composition of Functions

(f g)(x) = f(g(x)) is read f of g of x_, treat g(x) function first

Example.

For f(x) = 2x and g(x) = x + 4 evaluate: a) f(g(3)) b) Find the rule for f(g(x))

a) find g(3) = 3 + 4 = 7, next find f(7) = 2(7) = 14, therefore f(g(3)) = 14

b) f(g(x)) = f(x + 4) = 2(x + 4) = 2x + 8

 

Trigonometry

Sketch Sin & Cos Curves

|a| = amplitude

|b| = frequency

= period

To graph:

Find amp., freq., period, & set up chart by splitting period into quarters.

ex. period = p

x

trig function

y

0

 

 

 

 

 

 

 

 

p

 

 

Basic Tangent Curve

Find all quadrantal angles and 30, 60 and

all angles whose reference angles are 30 or 60

between 0 and 360.

 

Inverse Relations   Must find both angles q between 0° and 360° that has for it’s sine

ex.

 

 

 

Inverse Function     (Restricted Domains)       Sinx & Tanx     Cosx

Must find one angle q from the given restricted domians that has for it’s sine

ex.

(signs of each function)

I

II

III

IV

Sin

+

+

_

_

Cos

+

_

_

+

Tan

+

_

+

_

Exact Trig Values

0°

30°

45°

60°

90°

180°

270°

Sin A

0

1

0

-1

Cos A

1

0

-1

0

Tan A

0

1

undef.

0

undef.

Reciprocal Identities

Quotient Identities

Pythagorean Identities

Unit circle:

sin q = y = PQ

cos q = x = BQ

 

 

or (cos q, sin q)

 

 

 

 

 

 

Coterminal angles: Angles with the same terminal side ± 360k

Reference angles: q is an acute angle

Quadrant II 180 - q Quadrant III 180 + q Quadrant IV 360 - q

 

Finding trig functions on your calculator:

DD to DMS decimal degrees 2nd ANGLE to DMS

DMS to DD Degree 2nd ANGLE ° , minutes 2nd ANGLE ', second ALPHA " ENTER

 

Cofunctions: cofunctions of complements are equal.

sin A = cos B when A + B = 90°

tan A = cot B

sec A = csc B

Converting from degrees to radians:

Converting from radians to degrees: or replace p with 180°

 

Finding the measure of a central angle when the length of arc and radius are given:

or rq = s

Law of Cosines:

1. Use when given 2 sides and an included angle to find a 3rd side.

2. Use when given 3 sides of a triangle to find a angle.

Law of Sines:

1. Use when given 1 side and 2 angles to find a missing side.

2. The Ambiguous Case: a) Find measure asked for.

(unclear) b) State the number of distinct triangles

that could be drawn from given information.

Area of a Triangle. Area of D ABC =

Forces

When two forces are applied to a body, a 3rd force is produced, called a Resultant.

 

 

 

 

 

 

 

 

Circle Review

Central Angle: mΠ AOB = m                                 Diameter ^ to a chord: bisects the chord and arc.

 

Two tangent segments:                                 Tangent and a radius:                            

AP = PB                                                            radius ^ to the tangent

                                       

                  

Two chords:                                                              Tangent and a secant:

(AE)(EC) = (BE)(DE)                                                (tangent)2 = (secant)(external segement)

                                                                                    (CD)2 = (AC)(BC)

 

 

 

Two secants:                                                                                 Two chords:

(secant)(ext. segement) = (secant)(ext. segement)                            Π = ½ (the sum of the arcs formed by vertical angles)

(AD)(AB) = (AE)(AC)

Inscribed Angle: Π = ½ arc                                                         Angle formed by Chord & a tangent: Π = ½ arc

Angle formed at an external Point: Π = ½ (the difference of the two arcs formed)

Probability and Statistics

S The sum of related terms ex. means the sum of the squares of the integers from 1 to n.

Measures of Dispersion a number that indicates a variation within a set of data

Range -difference between highest and lowest value in data set

Mean- or average of your data

Standard Deviation =

To find, do STAT, Edit, type list into L1, STAT, CALC, 1-Var Stats, ENTER

 

Regressions - do STAT, Edit, type lists into L1 and L2, STAT, CALC, choose correct regression i. e. LinReg for a linear regression, ENTER, VARS, Y-VARS, Function, Y1, ENTER

LinReg(ax + b) – linear regressions QuadReg – Quadratic regressions

ExpReg – exponential regressions

Stat Plots- 2nd y = , ENTER, turn stat plot on, ZOOM, ZoomStat, enter

Normal Curve-see formula sheet

Probability of an exact event p = prob. of event q = prob. of not the event

n = # of trials r = # of sucesses

nCr prqn - r

Probability of at least r times sum of "r or more"

Probability of at most r times sum of "r or less"

Expanding a Binomial Use pascals triangle or formula below

(x + y)n =nC0 xn + nC1xn-1 y1 + nC2xn-2y2 + ... + nCn-1x1yn-1 + nCnx0yn

Finding the nth term of a Binomial Expansion nCr-1xn- (r-1)yr-1

Coordinate Rules of Transformations

Line Relections:

A reflection in the x-axis: rx-axis (x, y) = (x, - y)

A reflection in the y-axis: ry-axis (x, y) = (- x, y)

A reflection in the line y = x: ry=x (x, y) = (y, x)

A reflection in the line y = - x: ry=-x (x, y) = (- y, - x)

Point Reflections in the Origin:

A reflection in the origin, or a half-turn or a rotation of 180° : RO(x, y) = (- x, - y)

Rotations About the Origin: (always Counterclockwise, unless otherwise stated.)

Counterclockwise rotation of 90° about O: R90 (x, y) = (- y, x)

Counterclockwise rotation of 180° about O: R180 (x, y) = (- x, - y)

Counterclockwise rotation of 270° about O: R270 (x, y) = (y, - x)

Translation:

A translation of "a" units horizontally and "b" units vertically: Ta, b (x, y) = (x + a, y + b)

Dilation:

A dilation of k, where k > 0, and the origin is the center of dilation: Dk (x, y) = (kx, ky)

 

Coordinate Geometry Proofs

Proving an Isosceles Triangle

Find the distance of all three sides to show that two sides are equal

Proving an Isosceles Right Triangle

Find the distance of all three sides to show that two sides are equal, then show the Pythagorean theorem checks.

Proving a Parallelogram Either

1. Find the slope of all 4 sides and show both pairs of opposite sides are parallel because slopes are equal.

2. Find the distance of all 4 sides and show both pairs of opposite sides are equal in measure.

3. Show one pair of opposite sides are parallel (equal slopes) and congruent (equal distances).

Proving a Rectangle

Find the slopes of all 4 sides, show both pairs of opposite sides are parallel because slopes are equal, and consecutive sides have negative reciprocal slopes making them perpendicular.

Proving a Rhombus

Find the distance of all four sides and show they are all equal.

Proving a Square

Find the distance of all four sides and show they are all equal and find the slopes of at least two consecutive sides and show that they are negative reciprocal slopes making the sides perpendicular.

Proving a Trapezoid

Find the slopes of all four sides and show one pair of sides are parallel and the other pair of sides are not parallel.

Proving an Isosceles Trapezoid

Find the slopes of all four sides and show one pair of sides are parallel and the other pair of sides are not parallel and find the distance of the two non-parallel sides to show they are equal.

 

Fractions

Undefined Fraction:    A fraction  is undefined when b = 0 because 0 has no multiplicative inverse.
Multiplicative Inverse:    The product of any number and its multiplicative inverse is the identity 1.
Reducing Fractions To Lowest Terms
     1. Factor.
    2. Reduce any common factors using the cancellation process.
    3. Write reduced fraction.
Multiplying Fractions
    1. Factor each fraction.
    2. Reduce any common factors.
    3. Multiply any remaining factors.
    4. Write final fraction in reduced form.
Dividing Fractions
    1. To divide two fractions, multiply the dividend by the reciprocal of the divisor.
    2. Follow the rules of multiplication. Factor, Reduce, Multiply
Adding or Subtracting Fractions That Have the Same Denominator
    1. Add or subtract like terms in the numerator.
    2. Reduce sum or difference.
Adding or Subtracting Fractions That Have Different Denominators
    1. Find the lowest common denominator, write it down next to the problem.
    2. Get the lowest common denominator by multiplying the fraction(s) by "1".
    3. Add or subtract like terms in the numerator, reduce.
Solving Fractional Equations
    1. Add all fractions on the same side of the equation.
    2. Solve the resulting proportion. (Product of means equals Product of extremes)


Solving Equations

Absolute Value Equations                                        Example  9; 9; |3x – 4| + 5 = 6x

Algebraically                                                                                                                                 Graphically

1. Isolate the absolute value.

2. Write two equations by making the left + then –

3. Solve each equation

4. Check for extraneous solution(s).

Absolute Value Inequalities |2x + 5| + 2 < 7

Algebraically                                                                                                                                 Graphically

1. Isolate the absolute value.

2. Write two equations original

and 2nd by making right

negative and by changing

the inequality

3. Solve each equation

4. write inequality solution

sketch graph

Quadratic Inequalities

Algebraically                         x2 – 3x – 4 ³ 0                                                                                 Graphically

1. Solve for critical values.

2. Set up number line.

3. Find signs for intervals

4. Graph solution

 

 

 

 

 

Quadratic Equations

Algebraically x2 – 3x – 4 = 0

Factoring Quadratic Formula Graphically

9; 9; 9;

 

 

 

 

 

 

Radicals

Reducing Radicals:
1) Find the largest perfect square factor. ex.
2) Write the radical as a product of the factors.
3) Reduce the perfect square factor.

To add/subtract Radicals:
1) Reduce each radical. 2) Add/Subtract like radicals be adding coefficients.

To multiply/divide Radicals:
1) Multiply/divide coefficients 2) Multiply/divide radicals. 3) Simplify

Rationalizing the Denominator

1) Multiply numerator and denominator by the conjugate 2) Simplify

ex. 1.

Radical Equations  9;

Algebraically Graphically

1. Isolate the radical.

2. Square both sides

3. Solve each equation

4. Check for extraneous

solution(s).

Imaginary Numbers

Powers of i

Complex Numbers (a + bi)

Adding/Subtracting: Add/Subtract like terms ex. (2 + 3i) + (4 – 2i) = (6 + i)

Multiplying: Multiply and simplify ex.

Dividing: Rationalize the denominator by Multiplying the numerator and denominator by the conjugate and simplifying.

ex. 2

 

Geometry Proofs

Go to the web site for a great study guide

http://regentsprep.org/Regents/mathb/mathb.cfm#a1