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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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