New PDF release: Advanced Algebra II: Activities and Homework

By KennyFelder

Show description

Read Online or Download Advanced Algebra II: Activities and Homework PDF

Best algebra books

The Algebra of Logic by Louis Couturat PDF

In an admirably succinct shape, this quantity bargains a old view of the improvement of the calculus of good judgment, illustrating its attractiveness, symmetry, and straightforwardness from an algebraic standpoint. subject matters contain the rules of identification and the syllogism, the rules of simplification and composition; the legislation of tautology and of absorption; the distributive legislations and the legislation of duality, double negation, and contraposition; the formulation of De Morgan and Poretsky; Schröder's theorem; sums and items of services; resolution of equations concerning one and a number of other unknown amounts; the matter of Boole; Venn diagrams; tables of effects and factors; and formulation atypical to the calculus of propositions.

Additional resources for Advanced Algebra II: Activities and Homework

Example text

2, 4) (−1, 1) (0, 0) (1, 1) (2, 4) (4, −2) (1, −1) (0, 0) (1, 1) (4, 2) (2, π) (3, π) (4, π) (5, 1) (π, 2) (π, 3) (π, 4) (1, 5) ❊①❡r❝✐s❡ ✶✳✹✹ ▼❛❦❡ ✉♣ ❛ ❢✉♥❝t✐♦♥ ✐♥✈♦❧✈✐♥❣ ❛✳ ❲r✐t❡ t❤❡ s❝❡♥❛r✐♦✳ ♠✉s✐❝✳ ❨♦✉r ❞❡s❝r✐♣t✐♦♥ s❤♦✉❧❞ ❝❧❡❛r❧② t❡❧❧ ♠❡✖✐♥ ✇♦r❞s✖❤♦✇ ♦♥❡ ✈❛❧✉❡ ❞❡♣❡♥❞s ♦♥ ❛♥♦t❤❡r✳ ✷✶ ❜✳ ◆❛♠❡✱ ❛♥❞ ❝❧❡❛r❧② ❞❡s❝r✐❜❡✱ t✇♦ ✈❛r✐❛❜❧❡s✳ ■♥❞✐❝❛t❡ ✇❤✐❝❤ ✐s ❞❡♣❡♥❞❡♥t ❛♥❞ ✇❤✐❝❤ ✐s ✐♥❞❡♣❡♥❞❡♥t✳ ❝✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ s❤♦✇✐♥❣ ❤♦✇ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■❢ ②♦✉ ✇❡r❡ ❡①♣❧✐❝✐t ❡♥♦✉❣❤ ✐♥ ♣❛rts ✭❛✮ ❛♥❞ ✭❜✮✱ ■ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ②♦✉r ❛♥s✇❡r t♦ ♣❛rt ✭❝✮ ❜❡❢♦r❡ ■ r❡❛❞ ✐t✳ ❞✳ ❈❤♦♦s❡ ❛ s❛♠♣❧❡ ♥✉♠❜❡r t♦ s❤♦✇ ❤♦✇ ②♦✉r ❢✉♥❝t✐♦♥ ✇♦r❦s✳ ❊①♣❧❛✐♥ ✇❤❛t t❤❡ r❡s✉❧t ♠❡❛♥s✳ ❊①❡r❝✐s❡ ✶✳✹✺ ❍❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ ❢♦r ❛♥② ♥✉♠❜❡r x ✱ x2 − ✷✺ = (x + 5) (x − 5)✳ ❛✳ P❧✉❣ x = 3 ✐♥t♦ t❤❛t ❣❡♥❡r❛❧✐③❛t✐♦♥✱ ❛♥❞ s❡❡ ✐❢ ✐t ✇♦r❦s✳ ❜✳ ✷✵ × ✷✵ ✐s ✹✵✵✳ ●✐✈❡♥ t❤❛t✱ ❛♥❞ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥✱ ❝❛♥ ②♦✉ ✜♥❞ ✶✺ × ✷✺ ✇✐t❤♦✉t ❛ ❝❛❧❝✉❧❛t♦r❄ ✭❉♦♥✬t ❥✉st ❣✐✈❡ ♠❡ t❤❡ ❛♥s✇❡r✱ s❤♦✇ ❤♦✇ ②♦✉ ❣♦t ✐t✦✮ ❊①❡r❝✐s❡ ✶✳✹✻ ❆♠② ❤❛s st❛rt❡❞ ❛ ❝♦♠♣❛♥② s❡❧❧✐♥❣ ❝❛♥❞② ❜❛rs✳ ❊❛❝❤ ❞❛②✱ s❤❡ ❜✉②s ❝❛♥❞② ❜❛rs ❢r♦♠ t❤❡ ❝♦r♥❡r st♦r❡ ❛♥❞ s❡❧❧s t❤❡♠ t♦ st✉❞❡♥ts ❞✉r✐♥❣ ❧✉♥❝❤✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤ s❤♦✇s ❤❡r ♣r♦✜t ❡❛❝❤ ❞❛② ✐♥ ▼❛r❝❤✳ ❋✐❣✉r❡ ✶✳✾ ❛✳ ❖♥ ✇❤❛t ❞❛②s ❞✐❞ s❤❡ ❜r❡❛❦ ❡✈❡♥❄ ❜✳ ❖♥ ✇❤❛t ❞❛②s ❞✐❞ s❤❡ ❧♦s❡ ♠♦♥❡②❄ ❊①❡r❝✐s❡ ✶✳✹✼ ❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ s❤♦✇s t❤❡ ❣r❛♣❤ ♦❢ r✐❣❤t ❢♦r❡✈❡r✳ y= √ x✳ ❚❤❡ ❣r❛♣❤ st❛rts ❛t (0, 0) ❛♥❞ ♠♦✈❡s ✉♣ ❛♥❞ t♦ t❤❡ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✷ ❋✐❣✉r❡ ✶✳✶✵ ❛✳ ❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❣r❛♣❤❄ ❜✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❧♦♦❦s ❡①❛❝t❧② t❤❡ s❛♠❡✱ ❡①❝❡♣t t❤❛t ✐t st❛rts ❛t t❤❡ ♣♦✐♥t (−3, 1) ❛♥❞ ♠♦✈❡s ✉♣✲❛♥❞✲r✐❣❤t ❢r♦♠ t❤❡r❡✳ ❊①❡r❝✐s❡ ✶✳✹✽ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤ r❡♣r❡s❡♥ts t❤❡ ❣r❛♣❤ y = f (x)✳ ❋✐❣✉r❡ ✶✳✶✶ ❛✳ ❜✳ ❝✳ ❞✳ ❡✳ ■s ✐t ❛ ❢✉♥❝t✐♦♥❄ ❲❤② ♦r ✇❤② ♥♦t❄ ❲❤❛t ❛r❡ t❤❡ ③❡r♦s❄ ❋♦r ✇❤❛t x − values ✐s ✐t ♣♦s✐t✐✈❡❄ x − values ✐s ✐t ♥❡❣❛t✐✈❡❄ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ f (x)✳ ❖♥ ❋♦r ✇❤❛t ❇❡❧♦✇ ✐s t❤❛t s❛♠❡ ❣r❛♣❤✱ ❞r❛✇ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) − 2✳ ✷✸ ❋✐❣✉r❡ ✶✳✶✷ ❢✳ ❇❡❧♦✇ ✐s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ f (x)✳ ❖♥ t❤❛t s❛♠❡ ❣r❛♣❤✱ ❞r❛✇ t❤❡ ❣r❛♣❤ ♦❢ y = −f (x)✳ ❋✐❣✉r❡ ✶✳✶✸ ❊①tr❛ ❝r❡❞✐t✿ ❍❡r❡ ✐s ❛ ❝♦♦❧ tr✐❝❦ ❢♦r sq✉❛r✐♥❣ ❛ ❞✐✣❝✉❧t ♥✉♠❜❡r✱ ✐❢ t❤❡ ♥✉♠❜❡r ✐♠♠❡❞✐❛t❡❧② ❜❡❧♦✇ ✐t ✐s ❡❛s② t♦ sq✉❛r❡✳ 2 2 ❙✉♣♣♦s❡ ■ ✇❛♥t t♦ ✜♥❞ ✸✶ ✳ ❚❤❛t✬s ❤❛r❞✳ ❇✉t ✐t✬s ❡❛s② t♦ ✜♥❞ ✸✵ ✱ t❤❛t✬s ✾✵✵✳ ◆♦✇✱ ❤❡r❡ ❝♦♠❡s t❤❡ tr✐❝❦✿ ❛❞❞ ✸✵✱ ❛♥❞ t❤❡♥ ❛❞❞ ✸✶✳ ✾✵✵ + ✸✵ + ✸✶ = ✾✻✶✳ ❚❤❛t✬s t❤❡ ❛♥s✇❡r✦ ✸✶ 2 = ✾✻✶✳ ❛✳ ❯s❡ t❤✐s tr✐❝❦ t♦ ✜♥❞ ✹✶2 ✳ ✭❉♦♥✬t ❥✉st s❤♦✇ ♠❡ t❤❡ ❛♥s✇❡r✱ s❤♦✇ ♠❡ t❤❡ ✇♦r❦✦✮ ❜✳ ❲r✐t❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❣❡♥❡r❛❧✐③❛t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤✐s tr✐❝❦✳ ✶✷ ✶✳✶✷ ▲✐♥❡s ❊①❡r❝✐s❡ ✶✳✹✾ ❨♦✉ ❤❛✈❡ ✩✶✺✵ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ②❡❛r✳ ✭❈❛❧❧ t❤❛t ❞❛② ✏✵✑✳✮ ❊✈❡r② ❞❛② ②♦✉ ♠❛❦❡ ✩✸✳ ✶✷ ❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✶✾✶✶✸✴✶✳✶✴❃✳ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✹ ❛✳ ❜✳ ❝✳ ❞✳ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✶❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✹❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✶✵❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② n❄ ❚❤✐s ❣✐✈❡s ②♦✉ ❛ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥ ❢♦r ❤♦✇ ♠✉❝❤ ♠♦♥❡② ②♦✉ ❤❛✈❡ ♦♥ ❛♥② ❣✐✈❡♥ ❞❛②✳ ❡✳ ❍♦✇ ♠✉❝❤ ✐s t❤❛t ❢✉♥❝t✐♦♥ ❣♦✐♥❣ ✉♣ ❡✈❡r② ❞❛②❄ ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❢✳ ●r❛♣❤ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✵ ❨♦✉r ♣❛r❛❝❤✉t❡ ♦♣❡♥s ✇❤❡♥ ②♦✉ ❛r❡ ✷✱✵✵✵ ❢❡❡t ❛❜♦✈❡ t❤❡ ❣r♦✉♥❞✳ ✭❈❛❧❧ t❤✐s t✐♠❡ t = 0✳✮ ❚❤❡r❡❛❢t❡r✱ ②♦✉ ❢❛❧❧ ✸✵ ❢❡❡t ❡✈❡r② s❡❝♦♥❞✳ ✭◆♦t❡✿ ■ ❞♦♥✬t ❦♥♦✇ ❛♥②t❤✐♥❣ ❛❜♦✉t s❦②❞✐✈✐♥❣✱ s♦ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ ♣r♦❜❛❜❧② ♥♦t r❡❛❧✐st✐❝✦✮ ❛✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r ♦♥❡ s❡❝♦♥❞❄ ❜✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r t❡♥ s❡❝♦♥❞s❄ ❝✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r ✜❢t② s❡❝♦♥❞s❄ ❞✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r t s❡❝♦♥❞s❄ ❚❤✐s ❣✐✈❡s ②♦✉ ❛ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦r ②♦✉r ❤❡✐❣❤t✳ ❡✳ ❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ ②♦✉ t♦ ❤✐t t❤❡ ❣r♦✉♥❞❄ ❢✳ ❍♦✇ ♠✉❝❤ ❛❧t✐t✉❞❡ ❛r❡ ②♦✉ ❣❛✐♥✐♥❣ ❡✈❡r② s❡❝♦♥❞❄ ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❇❡❝❛✉s❡ ②♦✉ ❛r❡ ❢❛❧❧✐♥❣✱ ②♦✉ ❛r❡ ❛❝t✉❛❧❧② ❣❛✐♥✐♥❣ ♥❡❣❛t✐✈❡ ❛❧t✐t✉❞❡✱ s♦ t❤❡ s❧♦♣❡ ✐s ♥❡❣❛t✐✈❡✳ ❣✳ ●r❛♣❤ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✶ ▼❛❦❡ ✉♣ ❛ ✇♦r❞ ♣r♦❜❧❡♠ ❧✐❦❡ ❡①❡r❝✐s❡s ★✶ ❛♥❞ ★✷✳ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✱ ❛s ❛❧✇❛②s✳ ❇❡ ✈❡r② ❝❧❡❛r ❛❜♦✉t t❤❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ▼❛❦❡ s✉r❡ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡♠ ✐s ❧✐♥❡❛r✦ ❣❡♥❡r❛❧ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✷ ❈♦♠♣✉t❡ t❤❡ s❧♦♣❡ ♦❢ ❛ ❧✐♥❡ t❤❛t ❣♦❡s ❢r♦♠ (1, 3) t♦ (6, ✶✽)✳ ❊①❡r❝✐s❡ ✶✳✺✸ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠s✱ ✐♥❞✐❝❛t❡ r♦✉❣❤❧② ✇❤❛t t❤❡ s❧♦♣❡ ✐s✳ ❋✐❣✉r❡ ✶✳✶✹✿ ❛✳ ●✐✈❡ t❤❡ ✷✺ ❋✐❣✉r❡ ✶✳✶✺✿ ❜✳ ❋✐❣✉r❡ ✶✳✶✻✿ ❝✳ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✻ ❋✐❣✉r❡ ✶✳✶✼✿ ❞✳ ❋✐❣✉r❡ ✶✳✶✽✿ ❡✳ ✷✼ ❋✐❣✉r❡ ✶✳✶✾✿ ❢✳ ❊①❡r❝✐s❡ ✶✳✺✹ ◆♦✇✱ ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s✱ ❞r❛✇ ❛ ❧✐♥❡ ✇✐t❤ r♦✉❣❤❧② t❤❡ s❧♦♣❡ ✐♥❞✐❝❛t❡❞✳ ❋♦r ✐♥st❛♥❝❡✱ ♦♥ t❤❡ ✜rst ❧✐tt❧❡ ❣r❛♣❤✱ ❞r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ ✷✳ ❋✐❣✉r❡ ✶✳✷✵✿ ❜✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ m= −1 2 ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✽ ❋✐❣✉r❡ ✶✳✷✶✿ ❜✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ m= −1 2 ❋✐❣✉r❡ ✶✳✷✷✿ ❝✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ ♠ ❂ ✶ ❋♦r ♣r♦❜❧❡♠s ✼ ❛♥❞ ✽✱ • • • • • ❙♦❧✈❡ ❢♦r y✱ ❛♥❞ ♣✉t t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❢♦r♠ y = ♠① + b ✭.

2, 4) (−1, 1) (0, 0) (1, 1) (2, 4) (4, −2) (1, −1) (0, 0) (1, 1) (4, 2) (2, π) (3, π) (4, π) (5, 1) (π, 2) (π, 3) (π, 4) (1, 5) ❊①❡r❝✐s❡ ✶✳✹✹ ▼❛❦❡ ✉♣ ❛ ❢✉♥❝t✐♦♥ ✐♥✈♦❧✈✐♥❣ ❛✳ ❲r✐t❡ t❤❡ s❝❡♥❛r✐♦✳ ♠✉s✐❝✳ ❨♦✉r ❞❡s❝r✐♣t✐♦♥ s❤♦✉❧❞ ❝❧❡❛r❧② t❡❧❧ ♠❡✖✐♥ ✇♦r❞s✖❤♦✇ ♦♥❡ ✈❛❧✉❡ ❞❡♣❡♥❞s ♦♥ ❛♥♦t❤❡r✳ ✷✶ ❜✳ ◆❛♠❡✱ ❛♥❞ ❝❧❡❛r❧② ❞❡s❝r✐❜❡✱ t✇♦ ✈❛r✐❛❜❧❡s✳ ■♥❞✐❝❛t❡ ✇❤✐❝❤ ✐s ❞❡♣❡♥❞❡♥t ❛♥❞ ✇❤✐❝❤ ✐s ✐♥❞❡♣❡♥❞❡♥t✳ ❝✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ s❤♦✇✐♥❣ ❤♦✇ t❤❡ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡ ❞❡♣❡♥❞s ♦♥ t❤❡ ✐♥❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡✳ ■❢ ②♦✉ ✇❡r❡ ❡①♣❧✐❝✐t ❡♥♦✉❣❤ ✐♥ ♣❛rts ✭❛✮ ❛♥❞ ✭❜✮✱ ■ s❤♦✉❧❞ ❜❡ ❛❜❧❡ t♦ ♣r❡❞✐❝t ②♦✉r ❛♥s✇❡r t♦ ♣❛rt ✭❝✮ ❜❡❢♦r❡ ■ r❡❛❞ ✐t✳ ❞✳ ❈❤♦♦s❡ ❛ s❛♠♣❧❡ ♥✉♠❜❡r t♦ s❤♦✇ ❤♦✇ ②♦✉r ❢✉♥❝t✐♦♥ ✇♦r❦s✳ ❊①♣❧❛✐♥ ✇❤❛t t❤❡ r❡s✉❧t ♠❡❛♥s✳ ❊①❡r❝✐s❡ ✶✳✹✺ ❍❡r❡ ✐s ❛♥ ❛❧❣❡❜r❛✐❝ ❣❡♥❡r❛❧✐③❛t✐♦♥✿ ❢♦r ❛♥② ♥✉♠❜❡r x ✱ x2 − ✷✺ = (x + 5) (x − 5)✳ ❛✳ P❧✉❣ x = 3 ✐♥t♦ t❤❛t ❣❡♥❡r❛❧✐③❛t✐♦♥✱ ❛♥❞ s❡❡ ✐❢ ✐t ✇♦r❦s✳ ❜✳ ✷✵ × ✷✵ ✐s ✹✵✵✳ ●✐✈❡♥ t❤❛t✱ ❛♥❞ t❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥✱ ❝❛♥ ②♦✉ ✜♥❞ ✶✺ × ✷✺ ✇✐t❤♦✉t ❛ ❝❛❧❝✉❧❛t♦r❄ ✭❉♦♥✬t ❥✉st ❣✐✈❡ ♠❡ t❤❡ ❛♥s✇❡r✱ s❤♦✇ ❤♦✇ ②♦✉ ❣♦t ✐t✦✮ ❊①❡r❝✐s❡ ✶✳✹✻ ❆♠② ❤❛s st❛rt❡❞ ❛ ❝♦♠♣❛♥② s❡❧❧✐♥❣ ❝❛♥❞② ❜❛rs✳ ❊❛❝❤ ❞❛②✱ s❤❡ ❜✉②s ❝❛♥❞② ❜❛rs ❢r♦♠ t❤❡ ❝♦r♥❡r st♦r❡ ❛♥❞ s❡❧❧s t❤❡♠ t♦ st✉❞❡♥ts ❞✉r✐♥❣ ❧✉♥❝❤✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤ s❤♦✇s ❤❡r ♣r♦✜t ❡❛❝❤ ❞❛② ✐♥ ▼❛r❝❤✳ ❋✐❣✉r❡ ✶✳✾ ❛✳ ❖♥ ✇❤❛t ❞❛②s ❞✐❞ s❤❡ ❜r❡❛❦ ❡✈❡♥❄ ❜✳ ❖♥ ✇❤❛t ❞❛②s ❞✐❞ s❤❡ ❧♦s❡ ♠♦♥❡②❄ ❊①❡r❝✐s❡ ✶✳✹✼ ❚❤❡ ♣✐❝t✉r❡ ❜❡❧♦✇ s❤♦✇s t❤❡ ❣r❛♣❤ ♦❢ r✐❣❤t ❢♦r❡✈❡r✳ y= √ x✳ ❚❤❡ ❣r❛♣❤ st❛rts ❛t (0, 0) ❛♥❞ ♠♦✈❡s ✉♣ ❛♥❞ t♦ t❤❡ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✷ ❋✐❣✉r❡ ✶✳✶✵ ❛✳ ❲❤❛t ✐s t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❣r❛♣❤❄ ❜✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t❤❛t ❧♦♦❦s ❡①❛❝t❧② t❤❡ s❛♠❡✱ ❡①❝❡♣t t❤❛t ✐t st❛rts ❛t t❤❡ ♣♦✐♥t (−3, 1) ❛♥❞ ♠♦✈❡s ✉♣✲❛♥❞✲r✐❣❤t ❢r♦♠ t❤❡r❡✳ ❊①❡r❝✐s❡ ✶✳✹✽ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤ r❡♣r❡s❡♥ts t❤❡ ❣r❛♣❤ y = f (x)✳ ❋✐❣✉r❡ ✶✳✶✶ ❛✳ ❜✳ ❝✳ ❞✳ ❡✳ ■s ✐t ❛ ❢✉♥❝t✐♦♥❄ ❲❤② ♦r ✇❤② ♥♦t❄ ❲❤❛t ❛r❡ t❤❡ ③❡r♦s❄ ❋♦r ✇❤❛t x − values ✐s ✐t ♣♦s✐t✐✈❡❄ x − values ✐s ✐t ♥❡❣❛t✐✈❡❄ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ f (x)✳ ❖♥ ❋♦r ✇❤❛t ❇❡❧♦✇ ✐s t❤❛t s❛♠❡ ❣r❛♣❤✱ ❞r❛✇ t❤❡ ❣r❛♣❤ ♦❢ y = f (x) − 2✳ ✷✸ ❋✐❣✉r❡ ✶✳✶✷ ❢✳ ❇❡❧♦✇ ✐s t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ f (x)✳ ❖♥ t❤❛t s❛♠❡ ❣r❛♣❤✱ ❞r❛✇ t❤❡ ❣r❛♣❤ ♦❢ y = −f (x)✳ ❋✐❣✉r❡ ✶✳✶✸ ❊①tr❛ ❝r❡❞✐t✿ ❍❡r❡ ✐s ❛ ❝♦♦❧ tr✐❝❦ ❢♦r sq✉❛r✐♥❣ ❛ ❞✐✣❝✉❧t ♥✉♠❜❡r✱ ✐❢ t❤❡ ♥✉♠❜❡r ✐♠♠❡❞✐❛t❡❧② ❜❡❧♦✇ ✐t ✐s ❡❛s② t♦ sq✉❛r❡✳ 2 2 ❙✉♣♣♦s❡ ■ ✇❛♥t t♦ ✜♥❞ ✸✶ ✳ ❚❤❛t✬s ❤❛r❞✳ ❇✉t ✐t✬s ❡❛s② t♦ ✜♥❞ ✸✵ ✱ t❤❛t✬s ✾✵✵✳ ◆♦✇✱ ❤❡r❡ ❝♦♠❡s t❤❡ tr✐❝❦✿ ❛❞❞ ✸✵✱ ❛♥❞ t❤❡♥ ❛❞❞ ✸✶✳ ✾✵✵ + ✸✵ + ✸✶ = ✾✻✶✳ ❚❤❛t✬s t❤❡ ❛♥s✇❡r✦ ✸✶ 2 = ✾✻✶✳ ❛✳ ❯s❡ t❤✐s tr✐❝❦ t♦ ✜♥❞ ✹✶2 ✳ ✭❉♦♥✬t ❥✉st s❤♦✇ ♠❡ t❤❡ ❛♥s✇❡r✱ s❤♦✇ ♠❡ t❤❡ ✇♦r❦✦✮ ❜✳ ❲r✐t❡ t❤❡ ❛❧❣❡❜r❛✐❝ ❣❡♥❡r❛❧✐③❛t✐♦♥ t❤❛t r❡♣r❡s❡♥ts t❤✐s tr✐❝❦✳ ✶✷ ✶✳✶✷ ▲✐♥❡s ❊①❡r❝✐s❡ ✶✳✹✾ ❨♦✉ ❤❛✈❡ ✩✶✺✵ ❛t t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ②❡❛r✳ ✭❈❛❧❧ t❤❛t ❞❛② ✏✵✑✳✮ ❊✈❡r② ❞❛② ②♦✉ ♠❛❦❡ ✩✸✳ ✶✷ ❚❤✐s ❝♦♥t❡♥t ✐s ❛✈❛✐❧❛❜❧❡ ♦♥❧✐♥❡ ❛t ❁❤tt♣✿✴✴❝♥①✳♦r❣✴❝♦♥t❡♥t✴♠✶✾✶✶✸✴✶✳✶✴❃✳ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✹ ❛✳ ❜✳ ❝✳ ❞✳ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✶❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✹❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② ✶✵❄ ❍♦✇ ♠✉❝❤ ♠♦♥❡② ❞♦ ②♦✉ ❤❛✈❡ ♦♥ ❞❛② n❄ ❚❤✐s ❣✐✈❡s ②♦✉ ❛ ❣❡♥❡r❛❧ ❢✉♥❝t✐♦♥ ❢♦r ❤♦✇ ♠✉❝❤ ♠♦♥❡② ②♦✉ ❤❛✈❡ ♦♥ ❛♥② ❣✐✈❡♥ ❞❛②✳ ❡✳ ❍♦✇ ♠✉❝❤ ✐s t❤❛t ❢✉♥❝t✐♦♥ ❣♦✐♥❣ ✉♣ ❡✈❡r② ❞❛②❄ ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❢✳ ●r❛♣❤ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✵ ❨♦✉r ♣❛r❛❝❤✉t❡ ♦♣❡♥s ✇❤❡♥ ②♦✉ ❛r❡ ✷✱✵✵✵ ❢❡❡t ❛❜♦✈❡ t❤❡ ❣r♦✉♥❞✳ ✭❈❛❧❧ t❤✐s t✐♠❡ t = 0✳✮ ❚❤❡r❡❛❢t❡r✱ ②♦✉ ❢❛❧❧ ✸✵ ❢❡❡t ❡✈❡r② s❡❝♦♥❞✳ ✭◆♦t❡✿ ■ ❞♦♥✬t ❦♥♦✇ ❛♥②t❤✐♥❣ ❛❜♦✉t s❦②❞✐✈✐♥❣✱ s♦ t❤❡s❡ ♥✉♠❜❡rs ❛r❡ ♣r♦❜❛❜❧② ♥♦t r❡❛❧✐st✐❝✦✮ ❛✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r ♦♥❡ s❡❝♦♥❞❄ ❜✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r t❡♥ s❡❝♦♥❞s❄ ❝✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r ✜❢t② s❡❝♦♥❞s❄ ❞✳ ❍♦✇ ❤✐❣❤ ❛r❡ ②♦✉ ❛❢t❡r t s❡❝♦♥❞s❄ ❚❤✐s ❣✐✈❡s ②♦✉ ❛ ❣❡♥❡r❛❧ ❢♦r♠✉❧❛ ❢♦r ②♦✉r ❤❡✐❣❤t✳ ❡✳ ❍♦✇ ❧♦♥❣ ❞♦❡s ✐t t❛❦❡ ②♦✉ t♦ ❤✐t t❤❡ ❣r♦✉♥❞❄ ❢✳ ❍♦✇ ♠✉❝❤ ❛❧t✐t✉❞❡ ❛r❡ ②♦✉ ❣❛✐♥✐♥❣ ❡✈❡r② s❡❝♦♥❞❄ ❚❤✐s ✐s t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❇❡❝❛✉s❡ ②♦✉ ❛r❡ ❢❛❧❧✐♥❣✱ ②♦✉ ❛r❡ ❛❝t✉❛❧❧② ❣❛✐♥✐♥❣ ♥❡❣❛t✐✈❡ ❛❧t✐t✉❞❡✱ s♦ t❤❡ s❧♦♣❡ ✐s ♥❡❣❛t✐✈❡✳ ❣✳ ●r❛♣❤ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✶ ▼❛❦❡ ✉♣ ❛ ✇♦r❞ ♣r♦❜❧❡♠ ❧✐❦❡ ❡①❡r❝✐s❡s ★✶ ❛♥❞ ★✷✳ ❞❡♣❡♥❞❡♥t ✈❛r✐❛❜❧❡s✱ ❛s ❛❧✇❛②s✳ ❇❡ ✈❡r② ❝❧❡❛r ❛❜♦✉t t❤❡ ✐♥❞❡♣❡♥❞❡♥t ❛♥❞ ▼❛❦❡ s✉r❡ t❤❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡♠ ✐s ❧✐♥❡❛r✦ ❣❡♥❡r❛❧ ❡q✉❛t✐♦♥ ❛♥❞ t❤❡ s❧♦♣❡ ♦❢ t❤❡ ❧✐♥❡✳ ❊①❡r❝✐s❡ ✶✳✺✷ ❈♦♠♣✉t❡ t❤❡ s❧♦♣❡ ♦❢ ❛ ❧✐♥❡ t❤❛t ❣♦❡s ❢r♦♠ (1, 3) t♦ (6, ✶✽)✳ ❊①❡r❝✐s❡ ✶✳✺✸ ❋♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❞✐❛❣r❛♠s✱ ✐♥❞✐❝❛t❡ r♦✉❣❤❧② ✇❤❛t t❤❡ s❧♦♣❡ ✐s✳ ❋✐❣✉r❡ ✶✳✶✹✿ ❛✳ ●✐✈❡ t❤❡ ✷✺ ❋✐❣✉r❡ ✶✳✶✺✿ ❜✳ ❋✐❣✉r❡ ✶✳✶✻✿ ❝✳ ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✻ ❋✐❣✉r❡ ✶✳✶✼✿ ❞✳ ❋✐❣✉r❡ ✶✳✶✽✿ ❡✳ ✷✼ ❋✐❣✉r❡ ✶✳✶✾✿ ❢✳ ❊①❡r❝✐s❡ ✶✳✺✹ ◆♦✇✱ ❢♦r ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❣r❛♣❤s✱ ❞r❛✇ ❛ ❧✐♥❡ ✇✐t❤ r♦✉❣❤❧② t❤❡ s❧♦♣❡ ✐♥❞✐❝❛t❡❞✳ ❋♦r ✐♥st❛♥❝❡✱ ♦♥ t❤❡ ✜rst ❧✐tt❧❡ ❣r❛♣❤✱ ❞r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ ✷✳ ❋✐❣✉r❡ ✶✳✷✵✿ ❜✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ m= −1 2 ❈❍❆P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✽ ❋✐❣✉r❡ ✶✳✷✶✿ ❜✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ m= −1 2 ❋✐❣✉r❡ ✶✳✷✷✿ ❝✳ ❉r❛✇ ❛ ❧✐♥❡ ✇✐t❤ s❧♦♣❡ ♠ ❂ ✶ ❋♦r ♣r♦❜❧❡♠s ✼ ❛♥❞ ✽✱ • • • • • ❙♦❧✈❡ ❢♦r y✱ ❛♥❞ ♣✉t t❤❡ ❡q✉❛t✐♦♥ ✐♥ t❤❡ ❢♦r♠ y = ♠① + b ✭.

P❚❊❘ ✶✳ ❋❯◆❈❚■❖◆❙ ✷✵ ❜❛rs ✢♦❛t t❤r♦✉❣❤ t❤❡ ❛✐r ❛♥❞ ❧❛♥❞ ♦♥ t❤❡ t❡❛❝❤❡r✬s ❞❡s❦✳ ❆♥❞✱ ❛s q✉✐❝❦❧② ❛s s❤❡ ❛♣♣❡❛r❡❞✱ ❙❛❧❧② ✐s ❣♦♥❡ t♦ ❞♦ ♠♦r❡ ❣♦♦❞ ✐♥ t❤❡ ✇♦r❧❞✳ ▲❡t s r❡♣r❡s❡♥t t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ss✱ ❛♥❞ c r❡♣r❡s❡♥t t❤❡ t♦t❛❧ ♥✉♠❜❡r ♦❢ ❝❛♥❞② ❜❛rs ❞✐str✐❜✉t❡❞✳ ❚✇♦ ❢♦r ❡❛❝❤ st✉❞❡♥t✱ ❛♥❞ ✜✈❡ ❢♦r t❤❡ t❡❛❝❤❡r✳ ❛✳ ❲r✐t❡ ❛ ❢✉♥❝t✐♦♥ t♦ s❤♦✇ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ❙❛❧❧② ❣❛✈❡ ♦✉t✱ ❛s ❛ ❢✉♥❝t✐♦♥ ♦❢ t❤❡ ♥✉♠❜❡r ♦❢ st✉❞❡♥ts✳ c (s) =❴❴❴❴❴❴ ❜✳ ❯s❡ t❤❛t ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ t❤❡r❡ ✇❡r❡ ✷✵ st✉❞❡♥ts ✐♥ t❤❡ ❝❧❛ssr♦♦♠✱ ❤♦✇ ♠❛♥② ❝❛♥❞② ❜❛rs ✇❡r❡ ❞✐str✐❜✉t❡❞❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❝✳ ◆♦✇ ✉s❡ t❤❡ s❛♠❡ ❢✉♥❝t✐♦♥ t♦ ❛♥s✇❡r t❤❡ q✉❡st✐♦♥✿ ✐❢ ❙❛❧❧② ❞✐str✐❜✉t❡❞ ✸✺ ❝❛♥❞② ❜❛rs✱ ❤♦✇ ♠❛♥② st✉❞❡♥ts ✇❡r❡ ✐♥ t❤❡ ❝❧❛ss❄ ❋✐rst r❡♣r❡s❡♥t t❤❡ q✉❡st✐♦♥ ✐♥ ❢✉♥❝t✐♦♥❛❧ ♥♦t❛t✐♦♥✖t❤❡♥ ❛♥s✇❡r ✐t✳ ❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✵ ❚❤❡ ❢✉♥❝t✐♦♥ f (x) = ✐s ✏❙✉❜tr❛❝t t❤r❡❡✱ t❤❡♥ t❛❦❡ t❤❡ sq✉❛r❡ r♦♦t✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ f (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✶ ❚❤❡ ❢✉♥❝t✐♦♥ y (x) ✐s ✏●✐✈❡♥ ❛♥② ♥✉♠❜❡r✱ r❡t✉r♥ ✻✳✑ ❛✳ ❊①♣r❡ss t❤✐s ❢✉♥❝t✐♦♥ ❛❧❣❡❜r❛✐❝❛❧❧②✱ ✐♥st❡❛❞ ♦❢ ✐♥ ✇♦r❞s✿ y (x) =❴❴❴❴❴❴ ❜✳ ●✐✈❡ ❛♥② t❤r❡❡ ♣♦✐♥ts t❤❛t ❝♦✉❧❞ ❜❡ ❣❡♥❡r❛t❡❞ ❜② t❤✐s ❢✉♥❝t✐♦♥✿❴❴❴❴❴❴ ❝✳ ❲❤❛t x✲✈❛❧✉❡s ❛r❡ ✐♥ t❤❡ ❞♦♠❛✐♥ ♦❢ t❤✐s ❢✉♥❝t✐♦♥❄❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✷ z (x) = x2 − 6x + 9 ❛✳ z (−1) =❴❴❴❴❴❴ ❜✳ z (0) = ❴❴❴❴❴❴ ❝✳ z (1) =❴❴❴❴❴❴ ❞✳ z (3) =❴❴❴❴❴❴ ❡✳ z (x + 2) =❴❴❴❴❴❴ ❢✳ z (z (x)) =❴❴❴❴❴❴ ❊①❡r❝✐s❡ ✶✳✹✸ ❖❢ t❤❡ ❢♦❧❧♦✇✐♥❣ s❡ts ♦❢ ♥✉♠❜❡rs✱ ✐♥❞✐❝❛t❡ ✇❤✐❝❤ ♦♥❡s ❝♦✉❧❞ ♣♦ss✐❜❧② ❤❛✈❡ ❜❡❡♥ ❣❡♥❡r❛t❡❞ ❜② ❛ ❢✉♥❝t✐♦♥✳ ❆❧❧ ■ ♥❡❡❞ ✐s ❛ ✏❨❡s✑ ♦r ✏◆♦✑✖②♦✉ ❞♦♥✬t ❤❛✈❡ t♦ t❡❧❧ ♠❡ t❤❡ ❢✉♥❝t✐♦♥✦ ✭❇✉t ❣♦ ❛❤❡❛❞ ❛♥❞ ❞♦✱ ✐❢ ②♦✉ ✇❛♥t t♦.

Download PDF sample

Advanced Algebra II: Activities and Homework by KennyFelder


by Jeff
4.1

Rated 4.09 of 5 – based on 35 votes