SAT Subject Math Level 1 Facts & Formulas

Transcription

1 Numbers, Sequences, Factors Integers:..., -3, -2, -1, 0, 1, 2, 3,... Reals: integers plus fractions, decimals, and irrationals ( 2, 3, π, etc.) Order Of Operations: Aritmetic Sequences: PEMDAS (Parenteses / Exponents / Multiply / Divide / Add / Subtract) eac term is equal to te previous term plus d Sequence: t 1, t 1 + d, t 1 + 2d,... Te n t term is t n = t 1 + (n 1)d Number of integers from i n to i m = i m i n + 1 Sum of n terms S n = (n/2) (t 1 + t n ) (optional) Geometric Sequences: eac term is equal to te previous term times r Sequence: t 1, t 1 r, t 1 r 2,... Te n t term is t n = t 1 r n 1 Sum of n terms S n = t 1 (r n 1)/(r 1) (optional) Prime Factorization: break up a number into prime factors (2, 3, 5, 7, 11,... ) 200 = 4 50 = = 2 26 = Greatest Common Factor: multiply common prime factors 200 = = GCF(200, 60) = = 20 Least Common Multiple: ceck multiples of te largest number LCM(200, 60): 200 (no), 400 (no), 600 (yes!) Percentages: use te following formula to find part, wole, or percent part = percent 100 wole ttp:// pg. 1

2 Averages, Counting, Statistics, Probability average = sum of terms number of terms average speed = total distance total time Fundamental Counting Principle: sum = average (number of terms) mode = value in te list tat appears most often median = middle value in te list (wic must be sorted) Example: median of {3, 10, 9, 27, 50} = 10 Example: median of {3, 9, 10, 27} = (9 + 10)/2 = 9.5 If an event can appen in N ways, and anoter, independent event can appen in M ways, ten bot events togeter can appen in N M ways. (Extend tis for tree or more: N 1 N 2 N 3... ) Permutations and Combinations: Te number of permutations of n tings is n P n = n! Te number of permutations of n tings taken r at a time is n P r = n!/(n r)! Te number of combinations of n tings taken r at a time is n C r = n!/ ( (n r)! r! ) Probability: probability = number of desired outcomes number of total outcomes Te probability of two different events A and B bot appening is P (A and B) = P (A) P (B), as long as te events are independent (not mutually exclusive). If te probability of event A appening is P (A), ten te probability of event A not appening is P (not A) = 1 P (A). Logic (Optional): Te statement event A implies event B is logically te same as not event B implies not event A. However, event A implies event B is not logically te same as event B implies ttp:// pg. 2

3 event A. To see tis, try an example, suc as A = {it rains} and B = {te road is wet}. If it rains, ten te road gets wet (A B); alternatively, if te road is not wet, it didn t rain (not B not A). However, if te road is wet, it didn t necessarily rain (B A). Powers, Exponents, Roots x a x b = x a+b (x a ) b = x a b x 0 = 1 x a /x b = x a b (xy) a = x a y a xy = x y 1/x b = x b { ( 1) n +1, if n is even; = 1, if n is odd. Factoring, Solving If 0 < x < 1, ten 0 < x 3 < x 2 < x < x < 3 x < 1. (x + a)(x + b) = x 2 + (b + a)x + ab FOIL a 2 b 2 = (a + b)(a b) Difference Of Squares a 2 + 2ab + b 2 = (a + b)(a + b) a 2 2ab + b 2 = (a b)(a b) x 2 + (b + a)x + ab = (x + a)(x + b) Reverse FOIL You can use Reverse FOIL to factor a polynomial by tinking about two numbers a and b wic add to te number in front of te x, and wic multiply to give te constant. For example, to factor x 2 + 5x + 6, te numbers add to 5 and multiply to 6, i.e., a = 2 and b = 3, so tat x 2 + 5x + 6 = (x + 2)(x + 3). To solve a quadratic suc as x 2 +bx+c = 0, first factor te left side to get (x+a)(x+b) = 0, ten set eac part in parenteses equal to zero. E.g., x 2 + 4x + 3 = (x + 3)(x + 1) = 0 so tat x = 3 or x = 1. Te solution to te quadratic equation ax 2 + bx + c = 0 can always be found (if it exists) using te quadratic formula: x = b ± b 2 4ac. 2a Note tat if b 2 4ac < 0, ten tere is no solution to te equation. If b 2 4ac = 0, tere is exactly one solution, namely, x = b/2a. If b 2 4ac > 0, tere are two solutions to te equation. To solve two linear equations in x and y: use te first equation to substitute for a variable in te second. E.g., suppose x + y = 3 and 4x y = 2. Te first equation gives y = 3 x, so te second equation becomes 4x (3 x) = 2 5x 3 = 2 x = 1, y = 2. ttp:// pg. 3

4 Solving two linear equations in x and y is geometrically te same as finding were two lines intersect. In te example above, te lines intersect at te point (1, 2). Two parallel lines will ave no solution, and two overlapping lines will ave an infinite number of solutions. Functions A function is a rule to go from one number (x) to anoter number (y), usually written y = f(x). Te set of possible values of x is called te domain of f(), and te corresponding set of possible values of y is called te range of f(). For any given value of x, tere can only be one corresponding value y. Translations: Te grap of y = f(x ) + k is te translation of te grap of y = f(x) by (, k) units in te plane. Absolute value: x = { +x, if x 0; x, if x < 0. x < n n < x < n x > n x < n or x > n Parabolas: A parabola parallel to te y-axis is given by y = ax 2 + bx + c. If a > 0, te parabola opens up. If a < 0, te parabola opens down. Te y-intercept is c, and te x-coordinate of te vertex is x = b/2a. Compound Functions: A function can be applied directly to te y-value of anoter function. Tis is usually written wit one function inside te parenteses of anoter function. For example: f(g(x)) means: apply g to x first, ten apply f to te result g(f(x)) means: apply f to x first, ten apply g to te result f(x)g(x) means: apply f to x first, ten apply g to x, ten multiply te results For example, if f(x) = 3x 2 and g(x) = x 2, ten f(g(3)) = f(3 2 ) = f(9) = = 25. ttp:// pg. 4

5 Inverse Functions (Optional): Since a function f() is a rule to go from one number (x) to anoter number (y), an inverse function f 1 () can be defined as a rule to go from te number y back to te number x. In oter words, if y = f(x), ten x = f 1 (y). To get te inverse function, substitute y for f(x), solve for x in terms of y, and substitute f 1 (y) for x. For example, if f(x) = 2x + 6, ten x = (y 6)/2 so tat f 1 (y) = y/2 3. Note tat te function f(), given x = 1, returns y = 8, and tat f 1 (y), given y = 8, returns x = 1. Complex Numbers A complex number is of te form a + bi were i 2 = 1. Wen multiplying complex numbers, treat i just like any oter variable (letter), except remember to replace powers of i wit 1 or 1 as follows (te pattern repeats after te first four): i 0 = 1 i 1 = i i 2 = 1 i 3 = i i 4 = 1 i 5 = i i 6 = 1 i 7 = i For example, using FOIL and i 2 = 1: (1 + 3i)(5 2i) = 5 2i + 15i 6i 2 = i. Lines (Linear Functions) Consider te line tat goes troug points A(x 1, y 1 ) and B(x 2, y 2 ). Distance from A to B: Mid-point of te segment AB: Slope of te line: (x2 x 1 ) 2 + (y 2 y 1 ) 2 ( x1 + x 2 2, y ) 1 + y 2 2 y 2 y 1 = rise x 2 x 1 run Point-slope form: given te slope m and a point (x 1, y 1 ) on te line, te equation of te line is (y y 1 ) = m(x x 1 ). Slope-intercept form: given te slope m and te y-intercept b, ten te equation of te line is y = mx + b. To find te equation of te line given two points A(x 1, y 1 ) and B(x 2, y 2 ), calculate te slope m = (y 2 y 1 )/(x 2 x 1 ) and use te point-slope form. Parallel lines ave equal slopes. Perpendicular lines (i.e., tose tat make a 90 angle were tey intersect) ave negative reciprocal slopes: m 1 m 2 = 1. ttp:// pg. 5

6 a a b b a b b a a b m l b a Intersecting Lines Parallel Lines (l m) Intersecting lines: opposite angles are equal. Also, eac pair of angles along te same line add to 180. In te figure above, a + b = 180. Parallel lines: eigt angles are formed wen a line crosses two parallel lines. Te four big angles (a) are equal, and te four small angles (b) are equal. Triangles Rigt triangles: c a b 30 2x x 3 60 x x 2 45 x 45 x a 2 + b 2 = c 2 Special Rigt Triangles A good example of a rigt triangle is one wit a = 3, b = 4, and c = 5, also called a rigt triangle. Note tat multiples of tese numbers are also rigt triangles. For example, if you multiply tese numbers by 2, you get a = 6, b = 8, and c = 10 (6 8 10), wic is also a rigt triangle. All triangles: b Area = 1 2 b ttp:// pg. 6

7 Angles on te inside of any triangle add up to 180. Te lengt of one side of any triangle is always less tan te sum and more tan te difference of te lengts of te oter two sides. An exterior angle of any triangle is equal to te sum of te two remote interior angles. Oter important triangles: Equilateral: Tese triangles ave tree equal sides, and all tree angles are 60. Te area of an equilateral triangle is A = (side) 2 3/4. Isosceles: Similar: An isosceles triangle as two equal sides. Te base angles (te ones opposite te two sides) are equal (see te 45 triangle above). Two or more triangles are similar if tey ave te same sape. Te corresponding angles are equal, and te corresponding sides are in proportion. For example, te triangle and te triangle from before are similar since teir sides are in a ratio of 2 to 1. Trigonometry Referring to te figure below, tere are tree important functions wic are defined for angles in a rigt triangle: ypotenuse θ adjacent opposite sin θ = opposite ypotenuse SOH cos θ = adjacent ypotenuse CAH tan θ = opposite adjacent TOA (te last line above sows a mnemonic to remember tese functions: SOH-CAH-TOA ) An important relationsip to remember wic works for any angle θ is: sin 2 θ + cos 2 θ = 1. For example, if θ = 30, ten (refer to te Special Rigt Triangles figure) we ave sin 30 = 1/2, cos 30 = 3/2, so tat sin cos 2 30 = 1/4 + 3/4 = 1. ttp:// pg. 7

8 Circles (, k) r r n Arc Sector Area = πr 2 Circumference = 2πr Full circle = 360 Lengt Of Arc = (n /360 ) 2πr Area Of Sector = (n /360 ) πr 2 Equation of te circle (above left figure): (x ) 2 + (y k) 2 = r 2. Rectangles And Friends Rectangles and Parallelograms: Trapezoids: l w Rectangle Parallelogram (Square if l = w) (Rombus if l = w) Area = lw Area = l l w base 2 base 1 ( ) base1 + base 2 Area of trapezoid = 2 Polygons: Regular polygons are n-sided figures wit all sides equal and all angles equal. Te sum of te inside angles of an n-sided regular polygon is (n 2) 180. Te sum of te outside angles of an n-sided regular polygon is always 360. ttp:// pg. 8

9 Solids Te following five formulas for cones, speres, and pyramids are given in te beginning of te test booklet, so you don t ave to memorize tem, but you sould know ow to use tem. Volume of rigt circular cone wit radius r and eigt : V = 1 3 πr2 Lateral area of cone wit base circumference c and slant eigt l: S = 1 2 cl Volume of spere wit radius r: V = 4 3 πr3 Surface Area of spere wit radius r: S = 4πr 2 Volume of pyramid wit base area B and eigt : V = 1 3 B You sould know te volume formulas for te solids below. Te area of te rectangular solid is just te sum of te areas of its faces. Te area of te cylinder is te area of te circles on top and bottom (2πr 2 ) plus te area of te sides (2πr). r d w l Rectangular Solid Volume = lw Area = 2(lw + w + l) Rigt Cylinder Volume = πr 2 Area = 2πr(r + ) Te distance between opposite corners of a rectangular solid is: d = l 2 + w Te volume of a uniform solid is: V = (base area) eigt. ttp:// pg. 9