Essential Skills Chapter f ( x + h) f ( x ). Simplifying the difference quotient Section. h f ( x + h) f ( x ) Example: For f ( x) = 4x 4 x, find and simplify completely. h Answer: 4 8x 4 h. Finding the domain of a function Section. 4 Example: Find the domain of f ( x ) =. x 9 Answer: ( 9, ). Using functions as models to make predictions and draw conclusions Section. Example: If a rock falls from a height of 0 meters on the planet Jupiter, its height H (in meters) after x seconds is approximately H ( x) = 0 x. a. What is the height of the rock after second? Answer: 7 meters b. When is the height of the rock 0 meters? Answer: After approximately.88 seconds c. When does the rock strike the ground? Answer: After approximately.4 seconds 4. Finding information from the graph of a function Sections. and. Example: Use the graph of the function f below to answer the following: a. What is f ()? Answer: b. What is f () approximately? Answer:
c. What is the average rate of change of f on the interval [, ]? Answer: d. What is a local minimum value of f? Answer: 5 e. On what interval(s) is f increasing? Answer: [,] f. On what interval(s) is f decreasing? Answer: [ 5, ] [,] g. What is the domain of f? Answer: [ 5,] h. What is the range of f? Answer: [ 5,] i. For what values of x is f ( x ) <? Answer: { x 5 < x < } j. For what value of x is f ( x ) = 5? Answer: 5. Identifying the relationship between a function and its graph Section. x + Example: Let f ( x ) = x + 4, on the graph of f? Answer: yes a. Is the point ( ) 5 b. If x = 0, what is f ( x )? What corresponding point is on the graph of f? Answers:, ( 0, ) c. If f ( x ) =, what corresponding points are on the graph of f? d. Find the x and y intercepts of the graph of f. Answers: x intercept: none, y intercept: ( 0, ) 6. Finding the average rate of change of a function Section. Example: Find the average rate of change of the function f ( x) = x on the interval [ 7,9 ]. Answer: 4 7. Sketching graphs of basic functions Section.4 Example: Sketch the graph of x if x f ( x ) =. x + if x > Answer: 8. Sketching graphs of basic functions using transformations Section.5 Example: Sketch the graph of f x ( ) = ( x + )
Answer: 9. Constructing functions for modeling Section.6 Example: A rectangle with width x has a perimeter of 4 inches. a. Express the length of the rectangle l as a function of x. Answer: l( x) = x b. Find the domain of l. Express your answer in interval notation. Answer: (0,) c. Express the area of the rectangle A as a function of x. Answer: A( x) = x x
E s s e n tia l S kills C h a p t e r 4. I d e n t i fy i ng p r o p e r ti e s o f li n e ar fu n c t i o ns S e c tio n 4. E x a m p le : If f ( x ) = x +, a. D e te r m in e th e slo p e a nd y in te r c e p to f f. b. U s e th e s lo p e a nd y in te r c e p tt o g ra p h f. 4 c. D e te r m in e th e a v e r a g e ra te o f c h a n g e o f f o n th e in te r va l., 9 d. D e te r m in e w h e th e r f is in c r e as in g, d e c re a s in g, o r c o n s ta n t. A n s w e rs : a. s lo p e =, y i n t ecr e p t= b. sta rt a t (0, ), th e n g o d o w n th re e a n d rig h t tw o t o (, ) c. d. d e c re a s in g. U s i n g li ne a r fu n c t i o ns a s m o d e l s S e c tio n 4. E x a m p le : In 0 0, m a jo r le a g u e b a se b a l l s ig n e d a la b o r a g r e e m e n t w ith th e p la y e r s. In th is a g r e em e n t, a n y te a m w h o s e p ay ro ll e x c e e d s $ 8 m i l lio n s ta r tin g in 0 0 5 w i l lh a v e to p a y a lu x u ry ta x o f.5 % (fo r firs t tim e o ffe n s e s). T h e lin e a r fu n c tio n T ( p ) = 0. 5 ( p 8 ) d e sc rib e s th e lu x u ry ta x T o f a te a m w h o s e p ay ro ll is p (in m i l lio n s o f d o lla rs ). a. W h a t is th e im p l ie d d om a in o f th is fu n c tio n? b. W h a t is th e lu x u ry ta x fo r a te a m w h o s e p ay ro ll is is $ 6 0 m i l lo in? c. W h a t is th e p a y ro ll o f a te a m th a t p a y s a lu x u ry ta x o f $.7 m i l lo in? A n s w e rs : a. { p p 8 } b. $ 7. m i lio l n c. $ 8 0 m i lio l n. I d e n t i fy i ng p r o p e r ti e s o f q u a d r a ti c fu n c t i o ns S e c tio n 4. E x a m p le : L e t f ( x )= x 4 x. a. E x p re ss f in th e fo rm f ( x ) = a ( x h ) + k b. F in d th e v e r te x o f th e g ra p h o f f. c. F in d th e a x is o f s y m m e try o f th e g ra p h o f f. d. F in d th e x a n d y in te r c e p ts o f th e g ra p h o f f. e. W h e r e is f in c r e a sin g a n d d e c re a s in g? f. F in d th e d o m a in a n d ra n g e of f. g. D o e s f h a v e a m a x i m u m o r m in i m u m v a lu e? W h a t is th e m a x i m u m /m in i m u m v a lu e? h. S k e tc h th e g ra p h o f f.
A n s w e rs : a. ( 0, ) f x ( ) ( x ) = + b. (, ) c. x = d. x in te r c e p ts : n o n e, y in te r c e p t: e. in c r e a sin g o n (, ], d e c re a s in g o n [, ) f. d o m a in : (, ) (, ] g. m a x i m u m ;, ra n g e : h.. 4. F i n d i ng o p t i m a l v a lu e s o f q u a d r a ti c m o d e l s S e c tio n 4. E x a m p le : P a ra d is e T ra v e l A g e n c y s m o n th l y p ro fit P (in th o u s a n d s o f d o lla rs ) d e p e n d s o n th e am o u n t o f m o n e y x (in th o u s a n d s o f d o lla r s) s p e n t o n a d v e r tis in g p e r m o n th a c c o rd in g to th e ru le P ( x ) = 7 x ( x 4 ). W h a t is P a ra d is e s m a x im u m m o n th l y p ro fit? A n s w e r: $ 5,0 0 0 5. C o n s tr u c ti ng a n d u s i n g q u a d r a ti c m o d e l s S e c tio n 4.4 E x a m p le : A fa rm e r w ith 0 0 0 m e te r s o f fe n c in g w a n ts t o e n c lo s e a re c ta n g u la r p lo t th a t b o rd e r s o n a s tra ig h t h ig h w a y. If th e fa rm e r d o e s n o t fe n c e th e s id e a lo n g th e h ig h w a y, a. t o th e h ig h w a y ). b. W h a t is th e la r g e st a r e a th a t c a n b e e n clo s e d? Ep xre ss th e a re a, A, o f th e p lo t a s a fu n c tio n o f x (w h e r e x is th e s id e o f th e plo t p e r p e nd ic u la r A n s w e rs : a. A ( x ) 0 0 0 x x = b. 5 0 0,0 0 0 s q u a r e m e te r s
E s s e n tia l S kills C h a p t e r 5. G r a p h i ng p o ly n o m i a l fu n c t i o ns S e c tio n 5. E x a m p le : L e t f ( x ) = ( x ) ( x )( x + ). a. F in d th e x a n d y in te r c e p ts o f th e g ra p h o f f. b. D e te r m in e th e e nd b e h a v io r o f f. c. S k e tc h th e g ra p h o f f. A n s w e rs : a. x in te r c e p ts : (,0),(,0),(,0), y in te r c e p t: ( 0, ) b. a s x, y ; a s x, y c.. I d e n t i fy i ng p r o p e r ti e s o f p o ly n o m i al fu n c t i o ns S e c tio n 5. E x a m p le : U s e th e g ra p h o f a p oly n o m ia l fu n c tio n P ( x ) t o a n s w e r th e fo llo w in g q u e stio n s. a. W h a t is th e m in im u m d e g r e e of P ( x )? b. W h a t is th e sig n o f th e le a d in g c o e ffic ie n t o f P ( x )? c. Is th e d e g r e e of P ( x ) o d d o r e v e n? d. W h i c h o f th e fo llo w in g is m o s t lik e l y to b e th e fo rm u la fo r P ( x )? (c irc le o n e ) i) P ( x ) = x ( x + )( x )( x ) i i) i i) i iv ) P x x x x x ( ) = ( + ) ( )( ) P ( x ) = x ( x+ ) ( x )( x ) P ( x ) = x ( x + )( x ) ( x + ) A n s w e rs : a. 6 b. p o s itiv e c. e v e n d. iii
. G r a p h i ng r a ti o na l fu n c t i o ns S e c tio n s 5. a n d 5. x x E x a m p le : L e t R ( x ) =. x + a. F in d th e x a n d y in te r c e p ts o f th e g ra p h o f R. b. F in d th e e q u a tio n s o f a l l a sy m p to te s. c. S k e tc h th e g ra p h o f R. A n s w e rs : a. x in te r c e p ts : ( 4,0),(,0), y in te r c e p t: ( 0, ) b. v e r tic a l a sy m p to te : x =, h o riz o n ta l a sy m p to te : n o n e, ob liq u e a sy m p to te : y = x c. 4. S o lv i ng p o ly n o m i a l a n d r a ti o n a l i n e q u a li ti e s S e c tio n 5.4 x E x a m p le : S o lv e. x + x x, 0, A n s w e r: ( ] ( ] 5. F i n d i ng z e r o s o f p o ly n o m i als S e c tio n s 5.5 a n d 5.6 E x a m p le : F in d a l l th e z e r os o f P ( x ) = x 5 x + 6 x A n s w e r:, + i, i 6. W r i ti ng th e e q u a ti o n o f a p o ly n o m i a l u s i n g i ts z e r o s S e c t i o n 5.6 E x a m p le : W rite th e e q u a tio n o f th e fo u rth d e g r e e p o ly n o m i a l P ( x ) h a v in g i a z e ro, a z e ro o f m u ltip l ic ity, a n d a le a d in g c o e ffic ie n t o f. A n s w e r: P x x x x 4 ( ) = + 6 6 + 4
E s s e n tia l S kills C h a p t e r 6. F i n d i ng c o m p o s i t e fu n c t i o ns a n d t h e i rd o m a i n s S e c tio n 6. E x a m p le : F o r f ( x ) = a n d g ( x ) =, fin d ( f o g )( x ) a n d it s d o m a in. x + x x 5 A n s w e r: ( f o g )( x ) =, d o m a in = { } x 5 x x,. F i n d i ng i nv e rs e fu n c t i o ns S e c tio n 6. E x a m p le : F o r f ( x ) = x, fin d f ( x ). + x A n s w e r: f ( x ) = x. G r a p h i ng e x p o n e n ti a l fu n c t i o ns S e c tio n 6. x E x a m p le : S k e tc h th e g ra p h o f f ( x ) = 4 e. A n s w e r: 4. G r a p h i ng lo g a r i th m i c fu n c t i o ns S e c tio n 6.4 E x a m p le : S k e tc h th e g ra p h o f f ( x ) = lo g ( x + ). A n s w e r: 5. S i m p li fy i ng e x p r e s s i o ns i n v o lv i n g lo g a r i th m s S e c tio n 6.5 4 E x a m p le : W rite a s a s in g le lo g a rith m. 0 lo g x + lo g (4 x ) lo g 4 A n s w e r: 8 lo g ( x ) 6. S o lv i ng lo g a r i th m i c e q u a t i o n s S e c tio n 6.6 E x a m p le : S o lv e. lo g 5 x + lo g 5 ( x ) = A n s w e r: x = 5 7. S o lv i ng e x p o n e n ti a l e q u a t i o n s S e c tio n 6.6 x+ x E x a m p le : S o lv e. = 5
ln A n s w e r: x = ln ln 5 8. M o d e li ng u s i n g e x p o n e n ti a l fu n c t i o ns S e c tio n 6.8 E x a m p le : A p o p u la tio n o f b a c te ria ob e y s th e la w o f u n in h ib ite d g ro w th. If 6 0 0 b a c te ria a r e p re se n t in itia ll y a n d th e r e a r e 8 0 0 a fte r o n e ho u r, a. E x p re ss th e p o p u la tio n P a s a fu n c tio n o f tim e t. b. H o w lo n g w il l it b e u ntil th e p op u la tio n d o u b le s? (W rite a n e x a c t a n s w e r.) A n s w e r: a. 4 ln ( ) P ( t) = 6 0 0 e t b. ln ln ( ) h o u rs 4
E s s e n tia l S kills C h a p t e r 7. F i n d i ng th e c e n t e r a n d r a d i u s o f a c i rc le S e c tio n.4 E x a m p le : F in d th e c e n te r a n d ra d iu s o f th e circ le w ith e q u a tio n x y x y + 6 + 0 + 5 = 0. A n s w e r: c e n te r is (, 5 ), ra d iu s is. G r a p h i ng p a r a b o la s S e c tio n 7. E x a m p le : F o r th e p a ra b o la d efin e d b y th e e q u a tio n fo c u s, a n d d ire c trix a n d s k e tc h th e g ra p h. x x y 4 = 8 8, d e te rm in e th e v e rte x, A n s w e r: v e r te x is (, ), fo c u s is (,5 ), d ire c trix is y =. G r a p h i ng e l li ps e s S e c tio n 7. E x a m p le : F o r th e e ll ip s e d efin e d b y th e e q u a tio n x + y y + 9 = 0, d e te rm in e th e c e n te r, v e r tic e s, a n d fo c i a n d s k e tc h th e g ra p h. A n s w e r: c e n te r is (0, ), v e r tic e s a r e (, ) a n d (, ), fo c i a r e (, ) a n d (, ) 4. G r a p h i ng h y p e rb o la s S e c tio n 7.4 E x a m p le : F o r th e h y p e rb o la d efin e d b y th e e q u a tio n y x + 4 x 4 y = 0, d e te rm in e th e c e n te r, v e r tic e s, fo c i, tra n s v e r se ax is, a sy m p to te s a n d s k e tc h th e g ra p h.. A n s w e r: c e n te r is (, ), v e r tic e s a r e (, ) a n d (, ), fo c i a r e (, ) a n d (, + ), t ra n s v e r se a x is is x =, a sy m p t o te s a r e y = x a n d y = ( x )
5. U s i n g t he e q u a ti o ns o f c o n i c s to s o lv e a pp li ed p r o b le m s S e c tio n s 7., 7., 7.4 E x a m p le : A s a te ll ite d is h is s h a p e d lik e a p a r a bo lo id o f re v o lu tio n. T h e s ig n a ls th a t e m a n a te fro m a s a te ll ite s trik e th e s u rfa c e o f th e dis h a n d a r e re fle c te d t o a s in g le p o in t, w h e r e th e re c e iv e r is t o b e lo c a te d. f Ith e d is h is 0 fe e t a c ro s s a t its o p e n in g a n d 6 fe e t d e e p a t its c e n te r, a t w h a t p o s itio n s h o u ld th e re c e iv e r b e pla c e d? A n s w e r: It s h o u ld b e pla c e d fo o t a n d in c h fro m its b a se alo n g its a x is o f s y m m e try.
E s s e n tia l S kills C h a p t e r 8. S o lv i ng s y s te m s o f li ne a r e q u a t i o n s S e c tio n 8..5 x +. y =.7 E x a m p le : S o lv e..7 x. y =. A n s w e r: x =, y = 4. S o lv i ng s y s te m s o f n o n li n e ar e q u a t i o n s S e c tio n 8.6 E x a m p le : S o lv e. A n s w e r: ( 0, 0 ) x + y = 0 0 x y = 0 6,8 a n d ( ). U s i n g s y s te m s o f e q u a t i o n s to s o lv e a pp li e d p r o b le m s S e c tio n 8. E x a m p le : F in d re a l n u m b e r s a, b, a n d c s o th a t th e fu n c tio n y = a x + b x + c c o n ta in s th e p o in ts (,6),( ) A n s w e r:,8, a n d ( ) y x x 0, 4. 4 = + 4