Evet A: Fuctios ad Algebraic Maipulatio Factorig Square of a sum: ( a + b) = a + ab + b Square of a differece: ( a b) = a ab + b Differece of squares: a b = ( a b )(a + b ) Differece of cubes: a 3 b 3 = ( a b)(a + a b + b ) Sum of cubes: a 3 + b 3 = ( a + b)(a a b + b ) Sums, products, quotiets of ratioal expressios (x 3 + y 3 3 ) (x + y) x y Example: Express as the quotiet of terms ivolvig oly x ad y to the first power. Solvig equatios (icl. radical equatios) ivolvig these skills, but ultimately solvable by factorig/quadratic formula (o complex roots) Example: Fid both solutios to 3 13x + 37 3 13x 37 = 3. Ratioal expoets; simplifyig radical expressios a 0 = 1 a x = 1 a x a = b ( a) = a a m/ = a m a u a v = a u+v a u = a u v a v ( ab) u = a u u b ( a u ) v = a uv a u a ( b ) u = b u ab = a b a b = a b a 1/ = b a = b Fuctio otatio ad variatioal depedecies f (x) is a fuctio i x. That is, give some value for x, there is a uique correspodig value for f (x). Ex: if f (x) = x + 3, the f (5) = 5 + 3 = 8.
The fuctio g(f(x)), sometimes writte ( g f )(x), is a compositio of the fuctios f ad g. Ex: If f (x) = x ad g (x) = x 1, the f(g(x)) = f(x 1 ) = (x 1), or f (g(x)) = 4x 4 x + 1.
Evet B: Circular Figures ad Solids Cetral, iscribed, tagetial, ad exterior agles Cetral agle: A agle where the vertex is at the ceter of a circle ad its legs are radii of the circle. / AOC o the figure is a cetral agle. Iscribed agle: A agle formed from two chords that have a commo edpoit. / ABC o the figure is a iscribed agle. Theorem: A iscribed agle with legs AB ad C B is always half the agle of a cetral agle with legs AO ad C O. Power of a Poit Theorem (chords, secats, tagets) Secat: A lie that itersects a circle at two distict poits. Chord: A segmet betwee two poits o the circle. Taget: A lie that itersects a circle at oly oe poit. If chords AC ad BD itersect iside a circle at E (left): AE C E = BE D E. If secats AB ad ED itersect outside a circle at C (ceter): CB C A = CD C E. If a taget AB itersects a secat lie CD at B (right): AB = BC B D. Iterior ad exterior tagets of two circles:
Itercepted arcs Itercepted arc: The part of the circle that lies i betwee two lies that itersect it. I the figure, the arc is A C. To calculate the legth of a itercepted arc A C give the cetral agle / AOC (i degrees) that defies it (figure o the right), AOC l egth of arc AC = 360 / ( diameter of circle O). Area of circles, sectors, circular segmets Please tell me you kow the area of a circle: A = πr. For a sector with cetral agle θ, multiply the area of the circle by the proportio or percetage of the circle that the sector takes up: θ A = 360 πr. Circular segmet (figure at right): a regio bouded by a chord ad a arc of a circle. To fid the area of a segmet, draw the sector it lies iside, ad subtract the area of the triagle from the area of the etire sector. Cyliders, coes, spheres (volume & surface area) Cyliders: V = πrh ad S A = πr + πrh = πr(r + h ) 1 Coes: V = πr 3 h ad S A = πr + π rl = π r(r + l ) l = r + h is the slat height (see right) 4 Spheres: V = 3 πr 3 ad S A = 4πr
Evet C: Sequeces ad Series Sequeces: patters & recursive formulas, arithmetic/geometric seqs Arithmetic sequeces/series: Commo differece betwee terms is d. For a sequece havig terms, the geeral form is a, a + d, a + d, a + 3 d,..., a + ( 1 )d. Example: 5, 9, 13, 17, 1, 5 has = 6, a = 5, ad d = 4. The sum of a arithmetic series is the average of the first ad last terms multiplied with the umber of terms. Example: 1 5 9 3 3+3 6 = 78. 3 + 7 + 1 + 1 + 1 + = Geometric sequeces/series: Each term differs by a commo ratio r. For a sequece with terms, the geeral form of is a, ar, ar, ar 3,..., ar 1. Example: 3, 6, 1, 4, 48, 96 has = 6, a = 3, ad r =. The sum of a geometric series is: a + a r + ar +... + ar 1 ar a. 6 3 3 = r 1 19 Example: 3 + 6 + 1 + 4 + 48 + 96 = 1 = 3 1 = 189. A ifiite geometric sequece, a, ar, ar,... has a fiite sum if ad oly if 1 < r < 1. I that case, the sum is give by a + a r + ar +... = a. 1 r ar lim r a 1 (If you kow calculus, you ca derive this by takig ) Series: partial sums, formulas for 1 +... +, 1 +... +, 1 3 +... + 3.
Summatio/sigma otatio: If a k is some sequece, the k=0 a k = a 0 + a 1 + a +... + a ( + 1) k = 1 + +... + = k=1 k = 1 + +... + ( + 1) ( + 1) = 6 k=1 k 3 = 1 3 + 3 +... + 3 ( + 1) ( + 1) = 4 = ( ) k=1 * The sum of the first cubes is the square of the sum of the first itegers! Fuctio otatio; factorial otatio ad Biomial Theorem Defiitio:! = ( 1)( )... ()(1) Defiitio: ( ) (Biomial coefficiet; read chooser " ) r =! r!( r)! Biomial Theorem: The expasio of ( x + y), where is a positive iteger, is give by 0 0 + 1 ( x + y) 1 = ( ) x y ( ) x y 1 + ( ) x y 1 +... + ( ) x y ( ) x y 1 0 + ( ) x y. + 1 Example: ( a + b) 4 4 = ( )a4b 0 4 0 + ( ) a3b 1 4 1 + ( ) ab 4 + ( ) a1b 3 4 3 + ( ) a0 4 b 4, which simplifies to ( a + b) 4 = a 4 + 4a3 b + 6ab + 4ab b. 3 + 4 Pascal s Triagle is a triagular array of biomial coefficiets (left); each term is the sum of the two terms above it. To fid ( r ), go to row (row has as its secod umber, so the row with a sigle 1 is the 0th row) ad 5 across r terms. For example, ( ) is the 3rd term i row 5, so ( 5 ) = 10. Expadig ( a + b) gives a expressio with coefficiets followig the th row of Pascal s Triagle (for example, look at the expasio of ( a + b) 4 above: 1, 4, 6, 4, 1 ).
Evet D: Aalytic Geometry of the Coic Sectios Usig the stadard forms of equatios of the coic sectios Graphs, icludig the locatio of foci, directrices, ad asymptotes Use of properties of coics to solve applied problems, icl. max mi for parabolas
Circles Circles have the form ( x a) + ( y b) = r, where the ceter of the circle is ( a, b) ad the radius is r. Example: Compute the area of the circle described by x + y + x + 6 y + 3 = 0. Solutio: Completig the square, ( x + x + 1 ) + (y + 6 y + 9 ) + 3 = 1 + 9 ( x + 1) + ( y + 3) = 7 r = 7 ad the area of the circle is ( 7) π = 7π. Ellipse. Thus, Every poit P o a ellipse is such that the sum of the distace from each focus to P is costat. This sum is equal to r, where r is the legth of the semi major axis. (x h) (y k) Ellipses are of the form +, where is the axis alog the b = 1 a a x axis (major axis if a>b), b is the axis alog the y axis (major axis if b>a), ad ( h, k ) is the ceter of the ellipse. I a ellipse,, where is the distace from the ceter to either a b = c c focal poit.
Parabola Parabolas are either i the form y = a(x h) + k or x = a(y k) + h. Equatio y = a(x h) + k x = a(y k) + h Vertex ( h, k) ( h, k) Axis of Symmetry x = h y = k Focus ( h, k + 4a 1 ) ( h + 4a 1, k) Directrix y = k 4a 1 x = k 4a 1 Directio of Opeig Upward if a > 0, dowward if a < 0 Rightward if a > 0, Leftward if a < 0 Hyperbola
Hyperbolas have the form (x h) a b (y k) = 1. If the parabola is vertically orieted, flip ( x h) with ( y k ). I a hyperbola, a + b = c, where c is the distace from the ceter to a focus.