Xaverian Math Department



Functions – a relation in which for every x-value, there is one, and only one, y-value (vertical line test)

One-to-one function: for every y-value, there is one, and only one, x-value (horizontal line test)

Domain: all possible x-values

* denominator [pic] 0

* radicands should not be negative

Range: all possible y-values and x runs through the domain

Linear: (line) variables to the 1st power;

y = mx + b

Quadratic: (parabola) one variable squared;

y = ax2 + bx + c

To find roots:

* factor and follow with T-bar, OR

* use quadratic formula:

x = [pic]

axis of symmetry:

x = [pic] (substitute into equation to find min/max)

sum of roots: [pic] product of roots: [pic]

Nature of roots:

[pic]> 0, perfect square ( real, rational, unequal

[pic]> 0, not perf. square ( real, irrational, unequal

[pic]= 0, real, rational, equal

[pic]< 0, imaginary

Factoring:

1. GCF (one parentheses)

2. trinomial (reverse FOIL)

3. difference of perfect squares (conjugates)

Composition of functions: f(g(x)) or [pic]

( take 2nd function, g, and substitute for x in 1st function, f

To find the inverse function: switch x and y, solve for y

* to graph, reflect in y = x

[pic]

* if a = b, same signs ( circle

* if a [pic]b, same signs ( ellipse

* if a [pic]b, different signs ( hyperbola (function)

Equation of a Circle: [pic]

(h, k) = center r = radius

Direct variation: (line) [pic]( variables go in the same direction

Inverse variation:

(hyperbola)[pic], or, [pic]

( variables go in opposite directions

Trigonometry

tan[pic]; cot[pic]

csc[pic]; sec[pic]; cot[pic]

Graphs: for y = a sin (bx) + d, and y = a cos (bx) + d

[pic]; [pic]; d = vertical shift

period = [pic]

Pythagorean Triples:

(3, 4, 5); (5, 12, 13); (8, 15, 17)

Pythagorean Identities:

[pic]

[pic]

[pic]

Right triangle: SohCahToa

Special Right Triangles:

x x[pic] x[pic] 2x

x x

Degrees ( radians: multiply by [pic]

Radians ( degrees: multiply by [pic]

Arc length: [pic]

[pic]= central angle in radians

s = arc length

r = radius

Solving trig equations:

1. Convert equation so that only one type of function is being used

2. Solve for the function

3. Calculate inverse to solve for the angle

4. Use reference angles to find all angles that fall within the stated interval

(0,1)

II I

[pic] [pic]

S A

(-1,0) (1,0)

T C

[pic] [pic]

III IV

* reference angles are always

(0, -1) made with the x-axis!

| |30 |45 |60 |

|sin |½ |[pic] |[pic] |

|cos |[pic] |[pic] |1/2 |

|Tan |[pic] |1 |[pic] |

when to use…

Area of a Triangle – must have 2 sides and the [pic]between them

Law of Cosines – when you use 3 sides and 1 angle of a triangle

Law of Sines – when you use 2 sides and 2 opposite angles of a triangle; or when you are trying to determine the # of triangles that can be constructed

Parallelogram of forces: resultant force is the diagonal of a parallelogram formed by the two original forces

Geometry (Proofs)

Reflexive property: anything is [pic]to itself

Symmetric property: if A[pic]B, then B[pic]A

Transitive property: if A[pic]B and B[pic]C, then A[pic]C

Substitution property: equals may be substituted for each other

Complementary angles: add up to 90[pic]

Supplementary angles: add up to 180[pic]

Vertical angles are congruent

Proving [pic]are [pic]: SAS, SSS, ASA, HL, AAS ([pic] angles prove similarity only)

CPCTC: corresponding parts of [pic] [pic] are [pic]

Equation of a line: y = mx + b (m is slope; b is y-intercept)

Slope: [pic] or [pic]

[pic][pic][pic][pic]

* parallel lines have the same slope

* perpendicular lines have slopes that are negative reciprocals of one another

Circle Geometry

* 2 tangents from the same point are [pic]

* central angles [pic]intercepted arcs

* inscribed angles = ½ arc

* [pic] chords have [pic] arcs

* parallel chords “sandwich” [pic] arcs

* when a diameter is [pic]to a chord, it bisects the chord and its arcs

* chords equidistant from the center are [pic]

* when a tangent meets a radius, they form right [pic]’s

* when a tangent meets a chord, the [pic] = ½ the arc

* [pic] formed inside a circle = ½ (arc A + arc B)

* [pic] formed outside a circle = ½ (arc A – arc B)

* when 2 chords intersect, (piece)(piece) = (piece)(piece)

* when 2 secants meet at the same point,

(external)(entire) = (external)(entire)

* when a tangent and a secant meet at the same point, (tan)[pic]= (external)(entire)

Algebraic Fractions

To multiply:

1. factor

2. cancel out (top & bottom)

3. simplify whatever’s left

To divide:

1. keep, change, flip

2. follow rules of multiplication

To find LCD:

1. factor all denominators

2. place each factor in LCD once

To add/subtract:

1. find LCD

2. multiply ea. fraction by what’s missing (top/bottom)

3. add/subtract across the tops

4. reduce

Complex fractions:

1. find LCD

2. multiply tops of each fraction by LCD

3. cancel

4. simplify

Fractional equations:

1. find LCD

2. multiply tops only by LCD

3. cancel out denominators

4. solve

5. check (denominator [pic]0)

Solving[pic] equations:

1. isolate radical

2. square both sides

3. solve for variable

4. check in original equation

Solving [pic]equations:

1. isolate absolute value

2. set up two equations: one positive, one negative (if inequality, flip sign also!)

3. solve both equations

4. check in original equation

Solving quadratic inequalities:

1. solve for roots as in regular quadratic equation

2. for < or [pic], shade in

3. for > or [pic], shade out

Exponents

To multiply: add exponents ( [pic]

To divide: subtract ( [pic]

To raise a power to a power: multiply ( [pic]

* anything to the zero power = 1

Negative exponents: reciprocal ( [pic]

Fractional exponents: take root of denominator, then raise to power of numerator

Logarithms

[pic](exponential)[pic][pic] (logarithmic)

Product Rule: [pic]

Quotient Rule: [pic]

Power Rule: [pic]

To solve equations with variables in the exponent:

1. take the log of both sides

2. apply log rules

3. solve for variable

Imaginary & Complex #’s

[pic]

[pic]

* Simplify exponents by dividing by 4, use the remainder

Complex number: a + bi (a is real; bi is imaginary)

To graph: x-axis for real numbers (a); y-axis for imaginary numbers (b) ( connect to origin

Conjugate pair: binomials with same terms and one different sign ( (a + b)(a – b)

To rationalize denominators: multiply top and bottom by the denominator’s conjugate

ex: [pic] multiply by [pic]

* Do the same with radicals in the denominator!

Binomial Expansion

ex: (a + b)[pic]= [pic]

Probability

[pic]

n = # of trials (whole number)

r = # of successes (whole number)

p = probability of success on one trial

q = probability of failure on one trial

p + q = 1

Statistics

To find standard deviation and mean:

With frequency table:

1. enter scores into [pic] and frequency into[pic]

2. press STAT, CALC, 1-Var Stats, [pic],[pic], enter

With variables x, y:

1. enter data into[pic]and [pic]

2. press STAT, CALC, 2-Var Stats, enter (scroll down for [pic], etc)

Scatterplots:

1. go to STAT PLOT to activate Plot1

2. Zoom 9

Regression:

Linear: STAT, CALC, 4

Exponential: STAT, CALC, 0

( use equation format given, substitute values for coefficients a, b

Sigma notation: substitute values into rule from lower to upper limit and add all terms

ex: [pic]

Transformations

[pic]

[pic]

[pic]

[pic]

[pic]

To my students,

Take pride in everything you do. Be confident that you are capable of mastery. I have faith in you.. ( Ms. Chan

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[pic]

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