Grá co de uma Função Linear

✶

❋✉♥çõ❡s P♦❧✐♥♦♠✐❛✐s ❞❡ Pr✐♠❡✐r♦ ●r❛✉

R ❡♠ R❞❛❞❛ ♣♦r ✉♠❛ ❧❡✐ ❞❛

❢♦r♠❛

f(x) =ax+b,

♦♥❞❡a ❡b sã♦ ♥ú♠❡r♦s r❡❛✐s ❞❛❞♦s ❡a6= 0✳ ❖ ♥ú♠❡r♦a é ❝❤❛♠❛❞♦ ❞❡ ❝♦❡✜❝✐❡♥t❡ ❞❡x❡ ♦ ♥ú♠❡r♦ bé ❝❤❛♠❛❞♦ t❡r♠♦ ❝♦♥st❛♥t❡✳ ❱❡❥❛ ❛❧❣✉♥s ❡①❡♠♣❧♦s ❞❡ ❢✉♥çõ❡s ♣♦❧✐♥♦♠✐❛✐s ❞♦1◦ ❣r❛✉✿

✶✳ f(x) = 2x−3✱ ♦♥❞❡a= 2❡b=−3❀

✷✳ f(x) =−3x−4✱ ♦♥❞❡a=−3❡b=−4❀

✸✳ f(x) = 11x✱ ♦♥❞❡a= 11❡b= 0✳

❊①❡♠♣❧♦ ✶✳ ❖ ❝✉st♦ t♦t❛❧ ❞❡ ✉♠ ❢❛❜r✐❝❛♥t❡ ❝♦♥s✐st❡ ❡♠ 100 r❡❛✐s ✜①♦s ♠❛✐s 60 r❡❛✐s ❛ ❝❛❞❛ ♣❡ç❛ ❢❛❜r✐❝❛❞❛✳ ❈♦♥str✉❛ ✉♠❛ ❡①♣r❡ssã♦ ♣❛r❛ ❝❛❧❝✉❧❛r ♦ ❝✉st♦ ❡♠ ❢✉♥çã♦ ❞♦ ♥ú♠❡r♦ ❞❡ ♣❡ç❛s ♣r♦❞✉③✐❞❛s✳

❙❡❥❛ x♦ ♥ú♠❡r♦ ❞❡ ♣❡ç❛s ♣r♦❞✉③✐❞❛s✳ ♦ ❝✉st♦C(x)é ✐❣✉❛❧ ❛ 60x✱ ♣♦✐s ♣❛r❛ ❝❛❞❛ ♣❡ç❛ ♣r♦❞✉③✐❞❛ ❤á ✉♠ ❝✉st♦ ❞❡ 60 r❡❛✐s✱ ♠❛✐s ♦ ❝✉st♦ ✜①♦ ❞❡100 r❡❛✐s✿

❈✉st♦ t♦t❛❧=❈✉st♦ ❞❡ ❝❛❞❛ ♣❡ç❛ ×♥ú♠❡r♦ ❞❡ ♣❡ç❛s +❝✉st♦ ✜①♦ C(x) = 60x+ 100.

❖ ❝✉st♦ t♦t❛❧ ❛✉♠❡t❛ à t❛①❛ ❝♦♥st❛♥t❡ ❞❡60r❡❛✐s ♣♦r ♣❡ç❛ ♣r♦❞✉③✐❞❛✳ ❊ss❛ é ✉♠❛ ❝❛r❛❝t❡ríst✐❝❛ ❞❡ t♦❞❛s ❛s ❢✉♥çõ❡s ❧✐♥❡❛r❡s✿ ♦ ✈❛❧♦r ❞❛ ❢✉♥çã♦ ✈❛r✐❛ ❛ ✉♠❛ t❛①❛ ❝♦♥st❛♥t❡ ❡♠ r❡❧❛çã♦ à ✈❛r✐á✈❡❧ ✐♥❞❡♣❡♥❞❡♥t❡✳

●rá✜❝♦ ❞❡ ✉♠❛ ❋✉♥çã♦ ▲✐♥❡❛r

✷

❖ ❣rá✜❝♦ ❞❡ t♦❞❛ ❢✉♥çã♦ ❧✐♥❡❛r é ✉♠❛ ❧✐♥❤❛ r❡t❛ ✭♣♦r ✐ss♦ ♦ ♥♦♠❡ ❧✐♥❡❛r✮✳ ❙❡ ❛ r❡t❛ré ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦f(x) =ax+b✱ ❡♥tã♦ ❛ ❡q✉❛çã♦ y = ax+b é ❝❤❛♠❛❞❛ ❞❡ ❡q✉❛çã♦ ❞❛ r❡t❛ r✳ ➱ ♠✉✐t♦ ❢á❝✐❧ ❞❡s❡♥❤❛r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r✳ P♦❞❡♠♦s ❢❛③❡r ✐ss♦ ❞❡ ❞✉❛s ♠❛♥❡✐r❛s✳

✶✳ ❯♠❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r é ❡♥❝♦♥tr❛r ❞♦✐s ♣♦♥t♦s q✉❡ ♣❡rt❡♥❝❡♠ ❛♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦ ❡ ❡♠ s❡❣✉✐❞❛ ✉s❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ❞❛ ❣❡♦♠❡tr✐❛ ♣❧❛♥❛ q✉❡ ❞✐③✿

P♦r ❞♦✐s ♣♦♥t♦s ❞✐st✐♥t♦s ❞♦ ♣❧❛♥♦ ♣❛ss❛ ✉♠❛ ú♥✐❝❛ r❡t❛✳

❯♠❛ ✈❡③ ❝♦♥❤❡❝✐❞♦s ❞♦✐s ♣♦♥t♦s ♣❡rt❡❝❡♥t❡s ❛♦ ❣rá✜❝♦✱ ❜❛st❛ ❧✐❣❛r ❡ss❡ ♣♦♥t♦s ♣♦r ✉♠❛ r❡t❛ ♣❛r❛ ♦❜t❡r ♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳

❊①❡♠♣❧♦ ✷✳ ❙❡❥❛f(x) =x+ 1✳ P❛r❛ ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❡f ♣♦❞❡♠♦s ❡s❝♦❧❤❡r ❞♦✐s ✈❛❧♦r❡s ❞✐st✐♥t♦s q✉❛✐sq✉❡r ❞❡ x❡ ❝❛❧❝✉❧❛r ♦ ✈❛❧♦r ❝♦rr❡s♣♦♥❞❡♥t❡ ❞❡ f✿

f(0) = 1, f(1) = 2.

❊♠ s❡❣✉✐❞❛ ♠❛r❝❛♠♦s ♦s ♣♦♥t♦sp1= (0,1)❡p2= (1,2) ♥♦ ♣❧❛♥♦ ❝❛rt❡s✐❛♥♦✳ ❊ss❡s ♣♦♥t♦s ♣❡rt❡♥❝❡♠ ❛♦ ❣rá✜❝♦ ❞❡

f✱ q✉❡ ♣♦❞❡ s❡r ♦❜t✐❞♦ ❧✐❣❛♥❞♦ p1 ❡p2 ♣♦r ✉♠❛ ❧✐♥❤❛ r❡t❛✳

✷✳ ❖✉tr❛ ♠❛♥❡✐r❛ ❞❡ ❝♦♥str✉✐r ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r é ❝♦♥❤❡❝❡♥❞♦ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦x❡ ✉♠ ♣♦♥t♦ ♣♦r ♦♥❞❡ ❡❧❛ ♣❛ss❛✳ ❆ ✐♥❝❧✐♥❛çã♦m❞❡ ✉♠❛ r❡t❛ ❡♠ r❡❧❛çã♦ ❛♦ ❡✐①♦xé ❛ t❛♥❣❡♥t❡ ❞♦ â♥❣✉❧♦θ❡♥tr❡ ❡ss❛ r❡t❛ ❡ ♦ ❡✐①♦✿

❋✐❣✉r❛ ✶✿ ❆ ✐♥❝❧✐♥❛çã♦ ❞❡ ✉♠❛ r❡t❛ ém= tan(θ).

❯t✐❧✐③❛♥❞♦ ❛ ❞❡✜♥✐çã♦ ❞❡ t❛♥❣❡♥t❡ ❡ s❛❜❡♥❞♦ q✉❡ ❛ r❡t❛ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s(x1, y1)❡(x2, y2)t❡♠♦s q✉❡ ❛ ✐♥❝❧✐♥❛çã♦

mé ❞❛❞❛ ♣♦r

m= tan(θ) = ❝❛t❡t♦ ♦♣♦st♦ ❝❛t❡t♦ ❛❞❥❛❝❡♥t❡ =

y2−y1

x2−x1

= ∆y ∆x,

✸

❙❡ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ é f(x) =ax+b✱ ❡♥tã♦y2−y1= (ax2+b)−(ax1+b) =a(x2−x1)❡ ♣♦r ✐ss♦

m= tan(θ) = a(x2−x1) x2−x1

=a.

◆♦ ❡♥t❛♥t♦✱ s♦♠❡♥t❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡t❡r♠✐♥á−❧❛ ❝♦♠♣❧❡t❛♠❡♥t❡✳ ❖ ❞❡s❡♥❤♦ ❛❜❛✐①♦

♠♦str❛ ✈ár✐❛s r❡t❛s ❞✐❢❡r❡♥t❡s✱ t♦❞❛s ❡❧❛s ❝♦♠ ♦ ♠❡s♠♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r✳

P❛r❛ ❞❡t❡r♠✐♥❛r ❛ r❡t❛ ♣r❡❝✐s❛♠♦s ❝♦♥❤❡❝❡r t❛♠❜é♠ ♦ ✈❛❧♦r ❞❡b✳ ■ss♦ ♣♦❞❡ s❡r ❢❡✐t♦ s❡ ❝♦♥❤❡❝❡♠♦s ✉♠ ♣♦♥t♦ ♣♦r ♦♥❞❡ ❛ r❡t❛ ♣❛ss❛✳

❊①❡♠♣❧♦ ✸✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❝✉❥♦ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r−2 ❡ q✉❡ ♣❛ss❛ ♣❡❧♦ ♣♦♥t♦ (1,0)✳ ❊♥❝♦♥tr❡ ❛ ❡q✉❛çã♦ q✉❡ ❞❡✜♥❡f✳

❙❛❜❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r é ✐❣✉❛❧ ❛♦ ❝♦❡✜❝✐❡♥t❡ a✳ ❉❡ss❡ ♠♦❞♦✱ s❛❜❡♠♦s q✉❡ ❛ ❢✉♥çã♦ é ❞❛ ❢♦r♠❛f(x) =

✹

❆ ♠❛♥❡✐r❛ ♠❛✐s ❢á❝✐❧ ❞❡ ❡♥❝♦♥tr❛r ♦ ✈❛❧♦r ❞❡b é ✉s❛r ♦ ❢❛t♦ ❞❡ q✉❡f(0) =a×0 +b=b.■ss♦ s✐❣♥✐✜❝❛ q✉❡ ♦ ♣♦♥t♦ (0, b)é ♦ ♣♦♥t♦ ❞❡ ✐♥t❡rs❡çã♦ ❞♦ ❣rá✜❝♦ ❞❡f ❝♦♠ ♦ ❡✐①♦y✳

❊①❡♠♣❧♦ ✹✳ ❙❡❥❛ f ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❝✉❥♦ ❣rá✜❝♦ é ✉♠❛ r❡t❛ ❝♦♠ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r 1 ❡ ❝✉❥❛ ✐♥t❡rs❡çã♦ ❝♦♠ ♦ ❡✐①♦ y s❡❥❛ ♦ ♣♦♥t♦(0,4)✳

❙❛❜❡♠♦s q✉❡ ♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛r ❞❛ r❡t❛ é ✐❣✉❛❧ ❛ a❡ ❛ ✐♥t❡rs❡çã♦ ❞❛ r❡t❛ ❝♦♠ ♦ ❡✐①♦y é ♦ ♣♦♥t♦ (0, b)✳ ▲♦❣♦ ❛ ❢✉♥çã♦ é ❞❛❞❛ ♣❡❧❛ ❡①♣r❡ssã♦f(x) =x+ 4✳

❊①❡r❝í❝✐♦s

✶✳ ❊♥❝♦♥tr❡ ❛ ❡q✉❛çã♦ ❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❛ r❡t❛ q✉❡ ♣❛ss❛ ♣❡❧♦s ♣♦♥t♦s ❞❛❞♦s✿

✭❛✮ (0,1)❡(1,0)❀ ✭❜✮ (1,2)❡(2,4)❀

✭❝✮ (−1,3)❡(1,−3)❀

✭❞✮ (3,6)❡(2,−4)✳

✷✳ ❈♦♥str✉❛ ♦ ❣rá✜❝♦ ❞❛s r❡t❛s ❛❜❛✐①♦ ❡ ❞❡t❡r♠✐♥❡ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ ❝❛❞❛ ✉♠❛ ❞❡❧❛s✿ ✭❛✮ y= 2x+ 1❀

✭❜✮ y=−x+ 2❀ ✭❝✮ y= 3x+ 3❀ ✭❞✮ y=−2x❀

✭❡✮ y= 1❀ ✭❢✮ y=−2✳

✺

❈r❡s❝✐♠❡♥t♦ ❡ ❉❡❝r❡s❝✐♠❡♥t♦

❯♠❛ ❢✉♥çã♦ ❧✐♥❡❛r s❡rá s❡♠♣r❡ ❝r❡s❝❡♥t❡ ♦✉ ❞❡❝r❡s❝❡♥t❡✳ ❙❡✉ ❝♦♠♣♦rt❛♠❡♥t♦ ❡♠ r❡❧❛çã♦ ❛♦ ❝r❡s❝✐♠❡♥t♦ s❡rá ❞❛❞♦ ♣❡❧♦ ❝♦❡✜❝✐❡♥t❡ ❛♥❣✉❧❛ra✳ P❛r❛ ♣r♦✈❛r ❡ss❡ ❢❛t♦ ♣r❡❝✐s❛♠♦s ✉s❛r ✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ❞❛s ❞❡s✐❣✉❧❛❞❛s ❞♦ t✐♣♦m > n✳ ◗✉❛♥❞♦ ♠✉❧t✐♣❧✐❝❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦c >0❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥t✐♥✉❛ ✈❡r❞❛❞❡✐r❛✿

❙❡m > n❡c >0, ❡♥tã♦cm > cn.

❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦c <0❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡✈❡ s❡r ✐♥✈❡rt✐❞❛✿ ❙❡m > n❡c <0, ❡♥tã♦cm < cn.

✶✳ a >0 :◆❡ss❡ ❝❛s♦✱ ❛♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡x1> x2♣♦ra❡❧❛ s❡ ♠❛♥té♠ ❡ ♣♦rt❛♥t♦

x1> x2

⇒ ax1> ax2

⇒ ax1+b > ax2+b

⇒ f(x1)> f(x2)

❡ ♣♦rt❛♥t♦f é ✉♠❛ ❢✉♥çã♦ ❝r❡s❝❡♥t❡✳ ❙❡a >0✱ ♦ ❣rá✜❝♦ ❞❡f t❡♠ ❛ ❢♦r♠❛ ❛❜❛✐①♦✿

✷✳ a <0 :◆❡ss❡ ❝❛s♦✱ ❛♦ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡x1> x2♣♦ra❡❧❛ s❡ ✐♥✈❡rt❡ ❡ ♣♦rt❛♥t♦

x1> x2

⇒ ax1< ax2

⇒ ax1+b < ax2+b

✻

❡ ♣♦rt❛♥t♦f é ✉♠❛ ❢✉♥çã♦ ❞❡❝r❡s❝❡♥t❡✳ ❙❡a <0✱ ♦ ❣rá✜❝♦ ❞❡f t❡♠ ❛ ❢♦r♠❛ ❛❜❛✐①♦✿

❊①❡r❝í❝✐♦s

✶✳ ❉✐❣❛ s❡ ❛s ❢✉♥çõ❡s ❞❛❞❛s ❛❜❛✐①♦ sã♦ ❝r❡s❝❡♥t❡s✱ ❞❡❝r❡s❝❡♥t❡s ♦✉ ❝♦♥st❛♥t❡s✿ ✭❛✮ f(x) =x−1❀

✭❜✮ f(x) =−3❀

✭❝✮ f(x) =−2x+ 3❀ ✭❞✮ f(x) = 1❀

✭❡✮ f(x) = 3x−1✳

✷✳ ❙❛❜❡♥❞♦ q✉❡ ♦ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ é ✉♠❛ r❡t❛ ❝♦♠ ✐♥❝❧✐♥❛çã♦ ♣♦s✐t✐✈❛✱ ♦ q✉❡ ♣♦❞❡♠♦s ❞✐③❡r ❛ r❡s♣❡✐t♦ ❞♦ s❡✉ ❝r❡s❝✐♠❡♥t♦❄ ❊ s❡ ❡❧❛ ♣♦ss✉✐ ✐♥❝❧✐♥❛çã♦ ♥❡❣❛t✐✈❛❄ ◗✉❛♥❞♦ ❡❧❛ s❡rá ✉♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡❄

■♥t❡rs❡çã♦ ❝♦♠ ♦s ❡✐①♦s

❉❛❞❛ ✉♠❛ ❢✉♥çã♦y=f(x)✱ ♦s ✈❛❧♦r❡s ❞❡x♣❛r❛ ♦s q✉❛✐sf(x) = 0sã♦ ❝❤❛♠❛❞♦s r❛í③❡s ❞❡f✳ ◆♦ ❣rá✜❝♦ ❝❛rt❡s✐❛♥♦ ❞❛ ❢✉♥çã♦✱ ❛s r❛í③❡s sã♦ ❛❜s❝✐ss❛s ❞♦s ♣♦♥t♦s ♦♥❞❡ ♦ ❣rá✜❝♦ ❝♦rt❛ ♦ ❡✐①♦ ❤♦r✐③♦♥t❛❧✳ P❛r❛ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛rf(x) =ax+b t❡♠♦s✿

f(x) =ax+b= 0

⇒ ax=−b

⇒ x= −b a ▲♦❣♦ ❛ ❢✉♥çã♦f ❛❞♠✐t❡ ❛♣❡♥❛s ✉♠❛ r❛✐③✿ x= −b

a .

❖ ❣rá✜❝♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ♣♦❞❡ ✐♥t❡r❝❡♣t❛r ♦ ❡✐①♦ y ♥♦ ♠á①✐♠♦ ❡♠ ✉♠ ♣♦♥t♦✱ ❞❡✈✐❞♦ ❛♦ t❡st❡ ❞❛ r❡t❛ ✈❡rt✐❝❛❧✳ ❊ss❡ ♣♦♥t♦ é s❡♠♣r❡ ❞❛ ❢♦r♠❛(0, f(0))✳ P❛r❛ ❛ ❢✉♥çã♦ ❧✐♥❡❛rf(x) =ax+b✱ ❡ss❡ ♣♦♥t♦ é✱ ❝♦♠♦ ❥á ✈✐♠♦s✱ ♦ ♣♦♥t♦(0, b)✳ ❊①❡♠♣❧♦ ✺✳ ❊♥❝♦♥tr❡ ❛ ✐♥t❡rs❡çã♦ ❞❛ ❢✉♥çã♦f(x) =−2x+ 4 ❝♦♠ ♦s ❡✐①♦s ❝♦♦r❞❡♥❛❞♦s✳

✼

❊①❡r❝í❝✐♦s

❊♥❝♦♥tr❡ ❛ ✐♥t❡rs❡çã♦ ❡♥tr❡ ❛s r❡t❛s ❛❜❛✐①♦ ❡ ♦s ❡✐①♦s✿ ✶✳ f(x) =x❀

✷✳ f(x) =x−1❀

✸✳ f(x) =−3❀

✹✳ f(x) =−2x+ 3❀ ✺✳ f(x) = 1❀ ✻✳ f(x) = 3x−1✳

❊q✉❛çõ❡s ❡ ■♥❡q✉❛çõ❡s ❞❡ Pr✐♠❡✐r♦ ●r❛✉

❯♠❛ ❡q✉❛çã♦ ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉ é ✉♠❛ ❡q✉❛çã♦ ❞❛ ❢♦r♠❛ ax+b= 0.

❆s s♦❧✉çõ❡s ❞❡ss❛ ❡q✉❛çã♦ sã♦ ♦s ✈❛❧♦r❡s ❞❡ x♣❛r❛ ♦s q✉❛✐s ❡❧❛ é s❛t✐s❢❡✐t❛✳ ❊q✉❛çõ❡s ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉ ♣♦❞❡♠ s❡r ❢❛❝✐❧♠❡♥t❡ r❡s♦❧✈✐❞❛s✿

ax+b= 0 ⇒ ax=−b ⇒x= −b a .

❆ s♦❧✉çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉ é ❛ r❛✐③ ❞❛ ❢✉♥çã♦ ❧✐♥❡❛r f(x) =ax+b✳ ❯♠❛ ✐♥❡q✉❛çã♦ ❞❡ ♣r✐♠❡✐r♦ ❣r❛✉ é ✉♠❛ ✐♥❡q✉❛çã♦ ❞♦ t✐♣♦

ax+b >0, ax+b <0, ax+b ≥0 ♦✉ ax+b≤0.

❆ r❡s♦❧✉çã♦ ❞❡ ❡q✉❛çõ❡s ❞♦ ♣r✐♠❡✐r♦ ❣r❛✉ ❞❡♣❡♥❞❡ ❞♦ s✐♥❛❧ ❞♦ ❝♦❡✜❝✐❡♥t❡ a✳ ◆♦✈❛♠❡♥t❡ ✈❛♠♦s ✉s❛r ❛ ♣r♦♣r✐❡❞❛❞❡ ❜ás✐❝❛ ♣❛r❛ ❞❡s✐❣✉❛❧❞❛❞❡s✳ ◗✉❛♥❞♦ ♠✉❧t✐♣❧✐❝❛♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r ✉♠ ♥ú♠❡r♦ ♣♦s✐t✐✈♦c >0 ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❝♦♥t✐♥✉❛ ✈❡r❞❛❞❡✐r❛✿

❙❡m > n❡c >0, ❡♥tã♦cm > cn.

❙❡ ♠✉❧t✐♣❧✐❝❛r♠♦s ❛ ❞❡s✐❣✉❛❧❞❛❞❡ ♣♦r ✉♠ ♥ú♠❡r♦ ♥❡❣❛t✐✈♦c <0❛ ❞❡s✐❣✉❛❧❞❛❞❡ ❞❡✈❡ s❡r ✐♥✈❡rt✐❞❛✿ ❙❡m > n❡c <0, ❡♥tã♦cm < cn.

✶✳ a >0 :◆❡ss❡ ❝❛s♦✱ ❛♦ ❞✐✈✐❞✐r ♦✉ ♠✉❧t✐♣❧✐❝❛r ♣♦ra❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ ♠❛♥té♠ ax+b >0

⇒ ax >−b

⇒ x > −b a

❡ ♣♦rt❛♥t♦ ♦s ✈❛❧♦r❡s ❞❡xq✉❡ s❛t✐s❢❛③❡♠ ❛ ✐♥❡q✉❛çã♦ sã♦ ♦s ✈❛❧♦r❡s ❞❡xt❛✐s q✉❡ x > −b a .

✷✳ a <0 :◆❡ss❡ ❝❛s♦✱ ❛♦ ❞✐✈✐❞✐r ♦✉ ♠✉❧t✐♣❧✐❝❛r ♣♦ra❛ ❞❡s✐❣✉❛❧❞❛❞❡ s❡ ✐♥✈❡rt❡ ax+b >0

⇒ ax <−b

⇒ x < −b a

❡ ♣♦rt❛♥t♦ ♦s ✈❛❧♦r❡s ❞❡xq✉❡ s❛t✐s❢❛③❡♠ ❛ ✐♥❡q✉❛çã♦ sã♦ ♦s ✈❛❧♦r❡s ❞❡xt❛✐s q✉❡ x < −b a .

✽

❊①❡r❝í❝✐♦s

✶✳ ❊♥❝♦♥tr❡ ♦ ✈❛❧♦r ❞❡x✿ ✭❛✮ 2x+ 1 = 0❀ ✭❜✮ −x+ 3 = 4❀

✭❝✮ 3x= 0❀ ✭❞✮ x−1 = 2❀

✭❡✮ 5x+ 1 = 6✳

✷✳ ❊♥❝♦♥tr❡ ♦s ✈❛❧♦r❡s ❞❡xq✉❡ s❛t✐s❢❛③❡♠ ❛s ✐♥❡q✉❛çõ❡s ❛❜❛✐①♦✿ ✭❛✮ x+ 1>0❀

✭❜✮ x+ 3≤1❀

✭❝✮ 3x−1>0❀ ✭❞✮ −x−1>2❀

✭❡✮ 5x+ 1≥6✳

▲✐♠✐t❡s ❞❡ ❋✉♥çõ❡s ▲✐♥❡❛r❡s

❋✉♥çõ❡s ▲✐♥❡❛r❡s sã♦ ❢✉♥çõ❡s ♠✉✐t♦ ❜❡♠ ❝♦♠♣♦rt❛❞❛s✳ ❊♠ q✉❛❧q✉❡r ✈❛❧♦r ❞❡x0∈R✱ ❛ ♠❡❞✐❞❛ q✉❡ ♥♦s ❛♣r♦①✐♠❛♠♦s

❞❡x0 ♦ ✈❛❧♦r ❞❡ f(x) =ax+b s❡ ❛♣r♦①✐♠❛ ❞❡f(x0) =ax0+b✱ ❝♦♠♦ ♣♦❞❡♠♦s ✈❡r ❛ ♣❛rt✐r ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✿

❧♦❣♦ t❡♠♦s q✉❡✱ ♣❛r❛ q✉❛❧q✉❡r x0∈R

lim x→x0

f(x) =f(x0)

♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ t♦❞❛ ❢✉♥çã♦ ❧✐♥❡❛r é ❝♦♥tí♥✉❛✳

❖ ❝♦♠♣♦rt❛♠❡♥t♦ ❞❡ ✉♠❛ ❢✉♥çã♦ ❧✐♥❡❛r ❡♠ ±∞❞❡♣❡♥❞❡ ❞♦ s✐♥❛❧ ❞♦ ❝♦❡✜❝✐❡♥t❡a✳

✶✳ a >0✿ ◆❡ss❡ ❝❛s♦✱ q✉❛♥❞♦x❝r❡s❝❡✱ax t❛♠❜é♠ ❝r❡s❝❡✳ ▲♦❣♦ s❡xé s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ax+bs❡rá ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❡ ♣♦rt❛♥t♦

lim

x→∞f(x) =∞.

◗✉❛♥❞♦x❞❡❝r❡s❝❡ ❡ ✜❝❛ ♠✉✐t♦ ♥❡❣❛t✐✈♦✱axt❛♠❜é♠ ❞❡❝r❡s❝❡✳ ▲♦❣♦ s❡xé s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ ♥❡❣❛t✐✈♦ax+b s❡rá ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❡ ♥❡❣❛t✐✈♦ ❡ ♣♦rt❛♥t♦

lim

x→−∞f(x) =

−∞.

✾

✷✳ a <0✿ ◆❡ss❡ ❝❛s♦✱ q✉❛♥❞♦x❝r❡s❝❡✱ax❞❡❝r❡s❝❡✱ ✜❝❛♥❞♦ ❝❛❞❛ ✈❡③ ♠❛✐s ♥❡❣❛t✐✈♦✳ ▲♦❣♦ s❡xé s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ax+bs❡rá ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❡ ♥❡❣❛t✐✈♦ ❡ ♣♦rt❛♥t♦

lim

x→∞f(x) =−∞.

◗✉❛♥❞♦ x❞❡❝r❡s❝❡ ❡ ✜❝❛ ♠✉✐t♦ ♥❡❣❛t✐✈♦✱axs❡rá ✉♠ ♥ú♠❡r♦ ♠✉✐t♦ ❣r❛♥❞❡ ❡ ♣♦s✐t✐✈♦✳ ▲♦❣♦ s❡ xé s✉✜❝✐❡♥t❡♠❡♥t❡ ❣r❛♥❞❡ ❡ ♥❡❣❛t✐✈♦ ax+b s❡rá ✉♠ ♥ú♠❡r♦ ❣r❛♥❞❡ ❡ ♣♦s✐t✐✈♦ ❡ ♣♦rt❛♥t♦

lim

x→−∞f(x) =∞

❊ss❛s ♦❜s❡r✈❛çõ❡s t❛♠❜é♠ ♣♦❞❡♠ s❡r ♣r♦✈❛❞❛s ❛tr❛✈és ❞♦ ❣rá✜❝♦ ❞❛ ❢✉♥çã♦✳

❊①❡r❝í❝✐♦s

❈♦♥str✉❛ ♦ ❣rá✜❝♦ ❞❛s r❡t❛s ❛❜❛✐①♦ ❡ ❝❛❧❝✉❧❡ lim

x→0f(x), xlim→∞f(x) ❡ x→−∞lim f(x) :

✶✳ f(x) = 2x+ 1❀ ✷✳ f(x) =−x+ 2❀ ✸✳ f(x) = 3x+ 3❀ ✹✳ f(x) =−2x❀

✶✵

❊①❡r❝í❝✐♦s✿ ❆♣❧✐❝❛çõ❡s ❞❡ ❢✉♥çõ❡s ❧✐♥❡❛r❡s

✶✳ ❆ r❡❧❛çã♦ ❡♥tr❡ t❡♠♣❡r❛t✉r❛ ❡♠ ❣r❛✉s ❋❛❤r❡♥❤❡✐t(◦_F)❡ t❡♠♣❡r❛t✉r❛ ❡♠ ❣r❛✉s ❈❡❧s✐✉s ₍◦_C)é ❞❛❞❛ ♣❡❧❛ ❢ór♠✉❧❛✿

TF = 9

5TC+ 32.

❛✮ ❊s❜♦❝❡ ❛ r❡t❛ ❞❛❞❛ ♣❡❧❛ ❡q✉❛çã♦ ❛❝✐♠❛❀

❜✮ ◗✉❛❧ ❛ ✐♥❝❧✐♥❛çã♦ ❞❡ss❛ r❡t❛❄ ❖ q✉❡ ❡❧❛ r❡♣r❡s❡♥t❛❄

❝✮ ◗✉❛❧ é ❛ ✐♥t❡rs❡çã♦ ❞❡ss❛ r❡t❛ ❝♦♠ ♦ ❡✐①♦ y❄ ❖ q✉❡ ❡❧❛ r❡♣r❡s❡♥t❛❄

✷✳ ❙✉♣♦♥❤❛ ✉♠ ♦❜❥❡t♦ ❞❡ ❛rt❡ ❝♦♠♣r❛❞♦ ♣♦r ❘$50♠✐❧ ❛♣r❡s❡♥t❡ ✉♠❛ ❡①♣❡❝t❛t✐✈❛ ❞❡ s❡ ✈❛❧♦r✐③❛r à r❛③ã♦ ❝♦♥st❛♥t❡ ❞❡ ❘$5♠✐❧ ♣♦r ❛♥♦ ✭♣❡❧♦s ♣ró①✐♠♦s5❛♥♦s✮✳

❛✮ ❊s❝r❡✈❛ ❛ ❡q✉❛çã♦ ❞❛ r❡t❛ q✉❡ ❞❡s❝r❡✈❡ ♦ ✈❛❧♦r ❞♦ ♦❜❥❡t♦ ✭♣❛r❛ ♦s ♣ró①✐♠♦s 5❛♥♦s✮❀ ❜✮ ◗✉❛❧ s❡rá ♦ ✈❛❧♦r ❞♦ ♦❜❥❡t♦ ❞❡ ❛rt❡ três ❛♥♦s ❛♣ós ❞❛t❛ ❞❡ ❝♦♠♣r❛❄

✸✳ ❉❡s❞❡ ♦ í♥✐❝✐♦ ❞♦ ♠ês✱ ♦ r❡s❡r✈❛tór✐♦ ❞❡ á❣✉❛ ❞❡ ✉♠❛ ❝✐❞❛❞❡ ♣❡r❞❡ á❣✉❛ ❛ ✉♠❛ t❛①❛ ❝♦♥st❛♥t❡✳ ◆♦ ❞✐❛ ✶✷✱ ♦ r❡s❡r✈❛tór✐♦ t❡♠ ✷✵✵ ♠✐❧❤õ❡s ❞❡ ❧✐tr♦s❀ ♥♦ ❞✐❛ ✷✶✱ ❛♣❡♥❛s ✶✻✹ ♠✐❧❤õ❡s ❞❡ ❧✐tr♦s ❞❡ á❣✉❛✳

❛✮ ❊①♣r❡ss❡ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ á❣✉❛ ♥♦ r❡s❡r✈❛tór✐♦ ❡♠ ❢✉♥çã♦ ❞♦ t❡♠♣♦ ❡ ❝♦♥str✉❛ ♦ ❣rá✜❝♦ ❞❡ss❛ ❢✉♥çã♦❀ ❜✮◗✉❛♥t❛ á❣✉❛ ❤❛✈✐❛ ♥♦ r❡s❡r✈❛tór✐♦ ♥♦ ❞✐❛ ✽❄

✹✳ ❆ ❛❧t✉r❛ ♠é❞✐❛ H ❡♠ ❝❡♥tí♠❡tr♦s ❞❡ ✉♠❛ ❝r✐❛♥ç❛ ❞❡A❛♥♦s ❞❡ ✐❞❛❞❡ é ❞❛❞❛ ♣❡❧❛ ❢✉♥çã♦H = 6,5A+ 50✳ ❛✮ ◗✉❛❧ é ❛ ❛❧t✉r❛ ♠é❞✐❛ ❞❡ ✉♠❛ ❝r✐❛♥ç❛ ❞❡ 7❛♥♦s❄

❜✮◗✉❛❧ é ❛ ✐❞❛❞❡ ♣r♦✈á✈❡❧ ❞❡ ✉♠❛ ❝r✐❛♥ç❛ ❞❡150cm❄ ❝✮◗✉❛❧ é ❛ ❛❧t✉r❛ ♠é❞✐❛ ❞❡ ✉♠ r❡❝é♠✲♥❛s❝✐❞♦❄

❞✮◗✉❛❧ é ❛ ❛❧t✉r❛ ♠é❞✐❛ ❞❡ ✉♠ ❤♦♠❡♠ ❞❡20❛♥♦s s❡ ✉s❛r♠♦s ❡ss❛ ❡①♣r❡ssã♦❄ ❊ss❛ r❡s♣♦st❛ ♣❛r❡❝❡ r❛③♦á✈❡❧❄ ❖ q✉❡ ❡ss❡ r❡s✉❧t❛❞♦ ♥♦s ❞✐③ ❛ r❡s♣❡✐t♦ ❞❛ ❢✉♥çã♦H❄

✺✳ ❊♠ ❛❧❣✉♠❛s r❡❣✐õ❡s ❞♦ ♠✉♥❞♦✱ ♦❜s❡r✈♦✉✲s❡ q✉❡ ♦ ♥ú♠❡r♦N ❞❡ ♠♦rt❡s ♣♦r s❡♠❛♥❛ ❞❡♣❡♥❞❡ ❧✐♥❡❛r♠❡♥t❡ ❞❛ ❝♦♥❝❡♥✲ tr❛çã♦ x❞❡ ❞✐ó①✐❞♦ ❞❡ ❡♥①♦❢r❡ ♥♦ ❛r✳ ❙✉♣♦♥❤❛ q✉❡ t❡♥❤❛ ❤❛✈✐❞♦ ✾✼ ♠♦rt❡s q✉❛♥❞♦x= 100mg/m3

❡ ✶✶✵ ♠♦rt❡s q✉❛♥❞♦x= 500mg/m3

✳

❛✮ ◗✉❛❧ é ❛ ❡q✉❛çã♦ q✉❡ r❡❧❛❝✐♦♥❛N ❡x❄

❜✮ ◗✉❛♥t❛s ♠♦rt❡s ♦❝♦rr❡♠ ♣♦r s❡♠❛♥❛ s❡ x = 300 mg/m3_{❄ P❛r❛ q✉❛❧ ❝♦♥❝❡♥tr❛çã♦ ♦❝♦rr❡♠ ✶✵✵ ♠♦rt❡s ♣♦r}

s❡♠❛♥❛❄

✻✳ ❖ á❧❝♦♦❧ ❡tí❧✐❝♦ é ♠❡t❛❜♦❧✐③❛❞♦ ♣❡❧♦ ❝♦r♣♦ ❤✉♠❛♥♦ ❛ ✉♠❛ t❛①❛ ❝♦♥st❛♥t❡ ✭✐♥❞❡♣❡♥❞❡♥t❡ ❞❛ ❝♦♥❝❡♥tr❛çã♦✮✳ ❙✉♣♦♥❤❛ q✉❡ ❡ss❛ t❛①❛ s❡❥❛ ❞❡10mL♣♦r ❤♦r❛✳

❛✮ ◗✉❛♥t♦ t❡♠♣♦ é ♥❡❝❡ssár✐♦ ♣❛r❛ ❡❧✐♠✐♥❛r ♦s ❡❢❡✐t♦s ❞❡ ✉♠ ❧✐tr♦ ❞❡ ❝❡r✈❡❥❛ ❝♦♥t❡♥❞♦5%❞❡ á❧❝♦♦❧ ❡tí❧✐❝♦❄

❜✮ ❊①♣r❡ss❡ ♦ t❡♠♣♦T ♥❡❝❡ssár✐♦ ♣❛r❛ ♠❡t❛❜♦❧✐③❛r ♦ á❧❝♦♦❧ ❡tí❧✐❝♦ ❡♠ ❢✉♥çã♦ ❞❛ q✉❛♥t✐❞❛❞❡A❞❡ á❧❝♦♦❧ ❝♦♥s✉♠✐❞❛✳ ❝✮ ❉✐s❝✉t❛ ❞❡ q✉❡ ❢♦r♠❛ ❛ ❢✉♥çã♦ ♦❜t✐❞❛ ❛❝✐♠❛ ♣♦❞❡ s❡r ✉s❛❞❛ ♣❛r❛ ❞❡t❡r♠✐♥❛r ✉♠ ❧✐♠✐t❡ r❛③♦á✈❡❧ ♣❛r❛ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ á❧❝♦♦❧ ✐♥❣❡r✐❞❛ ❡♠ ✉♠❛ ❢❡st❛✳