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Math Midterm, TIP: 数学很有效的复习方法就是教会你朋友他/她一个不会的数学题/概念 或者 自己尝试制作思维导图,…
Math Midterm
Unit 2: quadratic equation
equation with one variable and the highest degree of 2 is quadratic equation
standard form: ax^2 + bx + c = 0 (a≠0)
solving quadratic equations(two common way)
identify the a, b, c in the quadratic equation, then apply them to the quadratic formula: [-b±√(b^2-4ac)] / 2a
cross multiplication: x^2 - 2x - 3 = 0 :arrow_right: (x-3)(x+1)=0 :arrow_right: x-3=0 or x+1 =0 :arrow_right: x=3 or x=-1
the discriminant (b^2 - 4ac) can identify the nature of solutions
if b^2 - 4ac > 0, there are two real roots
if b^2 - 4ac = 0, there are only one repeated root
if b^2 - 4ac < 0, there are no real roots
factoring the polynomials by finding the roots
ax^2 + bx + c = a(x - x1)(x - x2) (a≠0) <- x1 and x2 means two roots
韦达定理（挑战题可能用到哦）
：
x1 + x2 = -b/a
x1x2 = c/a
Application: calculate reducing / increasing rate -> original (100± rate)^turns = new
Unit 3: direct variation & inverse variation
function:
constant is a fixed value that does not change
in a function,
an input can only have one output
Domain
: set of all
input
values
Domain of the polynomial functions is R (for all real numbers)
Domain of the square root function √x is x≥0
Domain of a fractional function y = ƒ(x), denominator ≠ 0
Range: set of all output values
Direct variation function
relation between two quantities where the ratio of the two is equal to a constant value
y = kx (k≠0)
k is the constant
its graph can be determined by any two point that fits with the function, such as (0,0), (1,k)
when k>0, the line y=kx passes through the first and third quadrants; x increases, y increases; upward line
when k<0, the line y=kx passes through the second and fourth quadrants; x increases, y decreases; downward line
Inverse variation function
the product of two quantities is equal to a constant
xy=k (k≠0)
k is the constant
when k<0, the two curves pass through the second and fourth quadrants; x increases, y decreases;
when k>0, the two curves pass through the first and third quadrants; x increases, y increases;
two curves
do not touch the origin or intersect with the axes
, because then the denominator of y=k/x or x=k/y will be 0
Application: speed, time, distance (TBC)
Distance and Time
average speed = average distance / average time
Speed and Time
total distance traveled is the area of the graph
Unit 1: quadratic radical expressions
like radicals: radicals in simplest form which have the same radicands. (eg: 2√2 & 3√2 are like radicals)
the radicands contain no denominator
Methods of removing radicants
1- radication: 10/√5 = 10√5 / 5 = 2√5
2- multiplying the radicand in the denominator with its conjugate where we change the sign in the middle of two terms. (eg. the conjugate of (a+b) is (a-b): 1/(3-√2) = (3+√2) /(3-√2) (3+√2)= (3+√2) / 7
the radicands should be in the simplest form, having no perfect-square factors other than 1.
√(x+y)^2 = |x+y|
Unit 4: proposition & proof (TBC)
Proposition: a statement that can be judged true or false
true proposition (including theorem and axiom)
false proposition
If..... then...... (the part following if is hypothesis, the part following then is conclusion)
conditional: p->q
converse: q->p
inverse: ~p -> ~q
contrapositive: -q -> -p
Theorem: things that we have proven to be true
Axiom/ Postulate: things that we accept to be true without proving it
Geometrical theorems:
perpendicular bisector theorem (⊥ bisector theorem）
if the point is on the perpendicular bisector of a segment, then it is equidistant from the endpoint of the segment (If PM perpen. AB and MA = MB, then PA = PB)
converse
: if a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment (If PA = PB, OA=OB, then OP is the perpendicular bisector of AB)
angle bisector theorem
If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle (If QS bisects angle PQR, SP perpen. QP, SR perpen QR, then SP = SR)
converse
: If a point in the interior of an angle is equidistant from the sides of the angles, then the point is on the angle bisector( If SP perpen QP, SR perpen QR, and SP = SR, then QS bisects angle PQR)
Locus (abstract) is a set of consisting of all the points satifying specific conditions
E.g. A locus of 4cm from a point P is a circle with point P as the center and with a radius of 4cm (表述不准确，见谅）
Hypotenuse-Leg (HL) Theorem
conditions: two right triangles, the triangles have congruent hypotenuse, there is only one pair of congruent legs
written: HL (
,
)
Right angle
Rt ∆median to hypotenuse: If m∠ACB=90, then CD=AD=BD=AB/2
30ºRt∆
If m∠C=90, m∠A=30, then AB=2BC
converse
(Conv.:30ºRt∆): If m∠C=90, BC=AB/2, then m∠A=30
Rt ∆ acute∠s comp. : If ∠C=90, then m∠A + m∠B = 90
Pythagorean Theorem: If m∠C=90, then a^2 + b^2 = c^2
written: Pythagorean Theorem
converse
(Conv.:Pythagorean Theorem) If a^2 + b^2 = c^2, then m∠C=90
Distance Formula √[(x1 - x2)^2 + (y1-y2)^2]
TIP: 数学很有效的复习方法就是教会你朋友他/她一个不会的数学题/概念 或者 自己尝试制作思维导图
Representation for range or domain: { x | conditions for x}
perpen. is invented by Sunny, do not use it in your exams
