bestdiplomaprep.com Function Type General Function Form Original Function Horizontal Stretch y f ( x) y f (bx) Vertical Stretch Horizontal Translation Vertical Translation Combined Transformation y f ( x h) y f ( x) k y af b( x h) k h 0 Shift Right k 0 Shift Up h 0 Shift Left k 0 Shift Down Apply Transformations in BEDMAS Order (Stretches and reflections, then translations) ( x, y ) ( x, ay ) ( x, y ) ( x h, y ) ( x, y ) ( x, y k ) y af ( x) Vertical Stretch Horizontal Stretch By Factor 1 b By Factor a Wider: 0 b 1 No Effect Effect Study Smarter, Do Better Shorter: 0 a 1 Narrower: b 1 Taller: a 1 Reflect y axis : b 0 Reflect x axis : a 0 Mapped To ( x, y ) ( x, y ) ( x, y ) b x, y 1 1 ( x, y ) x h, ay k b y x y bx ya x y xh y x k y a b( x h) k y sin y sin b y a sin y sin h y sin k y a sin b( h) k y cos y cos b y a cos y cos h y cos k y a cos b( h) k Exponential y cx y c bx y ac x y c xh y cx k y ac b ( x h ) k Logarithmic y log x y log bx y a log x y log x h y log x k Radical Sinusiodal Polynomial Functions Remainder Theorem Form: f ( x) an x n an 1 x n 1 an 2 x n 2 a2 x 2 a1 x1 a0 Even Degree Odd Degree At least 1 to maximum of n x intercepts 0 to maximum of n x intercepts Domain x | x R Domain x | x R Range depends on max and min Range y | y R y -intercept ao y -intercept ao an 0 Qua dra nt II to I Polynomial has factor x - a x 1 into x 3 x 2 x 5 of multiplicity n 3 2 1 2 4 an 0 an 0 an 0 Qua dra nt III to IV Qua dra nt III to I Qua dra nt II to IV Multiplicity of Zero P( x) R Q( x) xa xa 1 1 3 2 5 1 2 4 No max or min Maximum and Minimum y a log b( x h) k 9 x3 3x 2 2 x 5 x 1 9 2 x 2x 4 x 1 repeated n times, x a is zero ex) y ( x 2) 2 x 2 is a zero multiplicty of 2 Angle Direction -'ve +'ve Logarithmic and Exponential Modeling Inverse of Function Radical Functions Domain of y Range of y f ( x) any value f ( x) 0 f ( x) 0 y f ( x) Denoted by: f ( x) any value f ( x) 0 y f ( x) and y f ( x) intersect Undefined at x 0 Swap x and y f ( x) is defined 0 f ( x) 1 y f ( x) above y f ( x) ( x, y ) ( y , x ) y f ( x) x f ( y ) f ( x) 1 y y f 1 ( x) f ( x) and y f ( x) intersect f ( x) 1 y Model Real Life Problems Using Final Quantity=initial quantity (change factor) number of changes y Final Quantity a initial quantity b change factor y a bx x number of changes f ( x) below y f ( x) at f ( x) 1 Solving Radical Functions Algebraically Factor Theorem 1) List Restrictions If x - a is a factor then P( a) 0 2)Isolate Radical and Square Both Sides If P(a) then x - a is a factor of P( x) Sine Law Pythagorean Theorem sin A sin B sin C a b c a 2 b2 c2 B Cosine Law a 2 b 2 c 2 2ab cos A 3)Find Roots of Equation (Solve for x) NOTE: Sine and Cosine Law 4)Check solutions in original equation True For Any Triangle Right Angle Triangle Only a c C A b Trig Identities Sum Double Angle Function Notation SOH CAH TOA sin A B sin A cos B cos A sin B sin 2 A 2sin A cos B h( x ) f ( x ) g ( x ) cos A B cos A cos B sin A sin B cos 2 A cos 2 A sin 2 A tan A tan B tan A B 1 tan A tan B cos 2 A 2 cos 2 A 1 h( x ) f ( x ) g ( x ) cos 2 A 1 2sin A h( x) ( f g )( x) opposite hypotenuse adjacent cos hypotenuse sin tan opposite adjacent 1 csc sin 1 sec cos 1 cot tan 2 Difference sin A B sin A cos B cos A sin B cos A B cos A cos B sin A sin B tan A B tan A tan B 1 tan A tan B Exponent Rules a a a b c bc a b c a bc tan 2 A 2 tan A 1 tan 2 A f ( x) g ( x) log c MN log c M log c N h( x ) M log c M log c N N log a M log c M log a C f h( x ) ( x ) g ( x ) 0 g log c ab a b c ac h( x) f ( x) g ( x) ( f g )( x) Log Rules y sin Trigonometry and Unit Circle h( x) ( f g )( x) h( x) f ( g ( x)) ( f g )( x) Max is 1 and Min is -1 Amplitude is 1 Period is 2 Period y -intercept is 0 2 360 or b b intercepts at - ,0, , 2 Domain | R Range y | 1 y 1, y R a b y cos 1 ab Max is 1 and Min is -1 Amplitude is 1 Period is 2 y -intercept is 1 Amplitude Max Value - Min Value 2 3 5 intercepts at - , , , 2 2 2 Domain | R Coterminal Angle a r 2 Range y | 1 y 1, y R CAST rule Binomial Theorem 360 n, n N 2 n, n N Coefficients of terms (n 1) row of Pascal's Triangle Permutations and Combinations n ! n (n 1) (n 2) 3 2 1 n objects with a of one kind that are identical and NOTE b of second kind that are identical n! can be arranged in ways a !b ! 0! 1 n Pr n Cr n! , n N (n r )! n! , n N (n r )!r ! n objects to choose from r number of items taken n Properties of Binomial Expansion of ( x y ) n N Order MATTERS Write Expansion in descending order of exponent of first term (In this case x) n objects to choose from Term number is k 1 r number of items taken Expansion contains n 1 terms Order DOESN'T matter Sum of exponents in any term n General Term Number tk 1 n Ck ( x) n k ( y ) k Copyright © 2013-2014 Fire Mountain Inc. 9/12/13