( ) 2 + k The vertex is ( h, k) ( )( x q) The x-intercepts are x = p and x = q.
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1 A Referece Sheet Number Sets Quadratic Fuctios Forms Form Equatio Stadard Form Vertex Form Itercept Form y ax + bx + c The x-coordiate of the vertex is x b a y a x h The axis of symmetry is x b a + k The vertex is ( h, k) ( x q) The x-itercepts are x p ad x q. y a x p The x-coordiate of the vertex is the average of the x-itercepts. Graphig Quadratics plot the vertex ad two other poits, oe to the left ad oe to the right of the vertex. Paret Fuctio: y x y ax + bx + c, a > 0 y ax + bx + c, a < 0 MINIMUM value at the vertex MAXIMUM value at the vertex Complex Numbers: a + bi (stadard form) Imagiary Numbers: 1 i Powers of i i 1 i i 1 i i i 4 1 Ratioalizig with Complex Numbers Ex: + i 7i *MULTIPLY the umerator ad deomiator by i + i i 7i i i + i 7i i i Ex: + i 7 i *MULTIPLY the umerator ad deomiator by the cojugate of the deomiator, 7+i + i 7 i 7+i 7+i 1+10i + i 49 i 1+10i i i
2 Completig the Square Use completig the square to Rewrite a quadratic fuctio from stadard form to vertex form. Solve quadratic equatios i stadard form. To complete the square:! b $ # & " a % y x +1x +10 y ( x + 6x ) y x + 6x + 9 y ( x + ) 8 4 Methods to Solve Quadratic Equatios Factorig Use whe ax + bx + c 0 ad ac has a set of factors that sum to b Square Roots - Use whe ax + c 0 or + k 0 a x h x 4x + 0 ( x 4x ) + 0 ( x 4x + 4) ( x ) +1 0 ( x ) 1 ( x ) 1 x i x ±i x ± i Completig the Square Use whe ax + bx + c 0 (*Coveiet whe b a is eve) Quadratic Formula Use whe ax + bx + c 0 (*Coveiet whe a, b,c are small.) x b ± The discrimiat: b 4ac *Use the discrimiat to determie the umber ad type of roots(solutios) of a quadratic equatio. If b 4ac > 0, Number of Solutios: distict Type of Solutios: real The graph of y ax + bx + c has two x-itercepts. Factorig If b 4ac 0, Number of Solutios: 1 repeated Type of Solutios: real The graph of y ax + bx + c has oe x-itercept. b 4ac a If b 4ac < 0, Number of Solutios: distict Type of Solutios: imagiary The graph of y ax + bx + c has NO x-itercepts.
3 Polyomial Fuctios Expoet Properties Ed Behavior Fidig Zeros Usig The Ratioal Zero Theorem Ratioal Expoets ad Radical Fuctios Ratioal Expoets ex: Evaluate. 4 / ( 4) Simplifyig Roots a m a m/ ( ) (easiest to take the root first!) Rule Whe is odd Whe is eve Example x x 64x y 4xy y x x 4 xy 4 x 8 x y 4 x ex: Simplify. a) b) 1 7 (look for perfect powers) 4 9 Fuctio Operatios Operatio Defiitio Additio f +g x Example: f ( x ) x, g ( x ) x + ( f + g) ( x ) x + x + x + x 1 ( f g) ( x ) x ( x + ) x x ( ) f ( x) + g ( x) ( f g) ( x ) f ( x ) g ( x ) ( fg) ( x ) f ( x ) g ( x ) ( fg) ( x ) ( x ) ( x + ) x + x x 6 Divisio!f$ f ( x) # &( x) g ( x) "g%!f$ x # &( x) x+ "g% Compositios ( f! g ) ( x ) f ( g ( x )) ( g! f ) ( x ) g ( f ( x )) ( f! g) ( x ) f ( x + ) ( x + ) x + 4x +1 ( g! f ) ( x ) g ( x ) x + x 1 Subtractio Multiplicatio
4 Iverse Fuctios Properties of Iverse Fuctios If f x ad g( x) are iverse fuctios the ( f! g) ( x) x AND( g! f )( x) x The graphs of f ( x) ad g( x) are reflectios about the lie y x The domai of f x is the rage of g( x). is the domai of g( x). cotais the poit (a, b) the g( x) cotais The rage of f x If f x the poit (b, a) To fid a Iverse: Ex: f x x switch x ad y x y + 4. solve for y y x 4. label appropriately f 1 ( x) x 4 Expoetial ad Logarithmic Fuctios Expoetial Fuctios: y ab x ( a 0, b > 0, b 1) b is the growth/decay factor If 0 < b <1, the y ab x represets expoetial decay. If b >1, the y ab x growth. represets expoetial! Ex: y 1 $ # & " 4 % x " Ex: y % $ ' # & x *otice the right side of the graph APPROACHES the asymptote Solvig Expoetial Equatios Cases Make the bases equal 9 x 7 x ( ) x x 4 6 x 1 x x 4 6x x x Logarithm Properties *otice the right side of the graph MOVES AWAY from the asymptote Ca t make the bases equal x+4 9 x+4 log log x+4 x + 4 log x 4 + log x.18 Quadratic Form x + x 1 0 ( x ) + x 1 0 Let u x u + u 1 0 ( u ) u 4 0 u, u 4 x, x 4 x 1
5 Solvig Logarithmic Equatios + 4 l x 4 l x 4 l x 4 e e x 4 e x 4 + e x 4.1 Compoud Iterest Iterest Compouded times per year A is the fial amout P is the iitial amout! A P# 1+ r t $ r is the iterest rate, i decimal & form " % t is the time i years is the umber of times the iterest is compouded per year *Goal ONE term o each side *CHECK FOR EXTRANEOUS SOLUTIONS logx + log x log x 10x log( 100) log log ( x 10 x) log x 10x 100 x 10x x x 0 0 ( x 10) x + 0 x 10, x x is extraeous, the aswer is x 10 Iterest Compouded Cotiuously A Pe rt A is the fial amout P is the iitial amout r is the iterest rate, i decimal form t is the time i years Domai Restrictios Fractios: y f x g x Domai: { x g( x) 0} Ex: y x + 9 x, { x x } Ex: y x x + 9, { x x R } Eve Roots: y f ( x), is eve Domai: { x f ( x) 0} Ex: y x + 7, { x x 7} Logarithms: y log b ( f ( x) ) Domai: { x f ( x) > 0} Ex: y l x + 4, { x x > 0}
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