Homework Section 3.1

MAT121 SECTION 3.1

HOMEWORK ANSWER KEY

10, 18, 26, 30, 35, 42

10. 𝒈 𝒙 = − 𝒙+ 𝟐 𝟐 + 𝟒

a. The parabola opens down.

b. The vertex is −2, 4 .

c. The x-intercepts are 0,0 and (−4, 0).

− 𝑥 + 2 ! + 4 = 0

− 𝑥 + 2 ! = −4

𝑥 + 2 ! = 4

𝑥 + 2 = 4

𝑥 = −2± 2

𝑥 = 0,−4

d. The y-intercept is (0,0).

𝑔 0 = − 0+ 2 ! + 4 = 0

e. Sketch the graph.

f. The axis of symmetry is 𝑥 = −2.

g. The function has a maximum value of 4.

h. Domain: (−∞,∞) Range: (− ∞, 4]

18. 𝒈 𝒙 = 𝒙𝟐 + 𝟖𝒙+ 𝟕

a. Vertex Form: 𝑥 + 4 ! − 9

(𝑥! + 8𝑥 + 16)+ 7− 16

b. The vertex is −4,−9 .

c. The x-intercepts are −1,0 and (−7, 0).

𝑥 + 1 𝑥 + 7 = 0

d. The y-intercept is (0,7).

e. Sketch the graph.

f. The axis of symmetry is 𝑥 = −4

g. The function has a minimum value of −9.

h. Domain: (−∞,∞) Range: [−9,∞)

Homework Section 3.1

26. 𝒋 𝒕 = − 𝟏𝟒𝒕𝟐 + 𝟏𝟎𝒕− 𝟓

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = −𝑏2𝑎 =

−10

2 − 14=−10

− 12= 20

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = 𝑗 20 = −14 20 ! + 10 20 − 5 = −

14 400 + 200− 5 = 95

𝑉𝑒𝑟𝑡𝑒𝑥 = (20, 95)

30. 𝒌 𝒙 = 𝟐𝒙𝟐 − 𝟏𝟎𝒙− 𝟓

a. The parabola opens up.

b. The vertex is 2.5,−17.5 .

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = −𝑏2𝑎 =

− −102 2 =

52 = 2.5

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = 𝑘52 = 2

52

!

− 1052 − 5 = −17.5

c. The x-intercepts are 5.46, 0 and (−.46, 0).

Via DQUAD: 𝑥 = !!± !"

!

d. The y-intercept is (0,−5).

𝑘 0 = 2 0 ! − 10 0 − 5 = −5

e. Sketch the graph.

f. The axis of symmetry is 𝑥 = 2.5.

g. The function has a minimum value of -17.5.

h. Domain: (−∞,∞) Range: [−17.5, ∞)

35. 𝒉 𝒙 = −𝟎.𝟎𝟐𝟔𝒙𝟐 + 𝟎.𝟓𝟕𝟔𝒙+ 𝟑

a. The maximum height occurs at the vertex. The maximum height is 11.1 meters.

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = −𝑏2𝑎 =

−0.5762 −0.026 = 11.1

b. The maximum height is 6.2 meters.

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = ℎ 11.1 = −0.026 11.1 + 0.576 11.1 + 3 = 6.2

c. The firefighter is 14 meters from the house.

−0.026𝑥! + 0.576𝑥 + 3 = 6 simplified to −0.026𝑥! + 0.576𝑥 − 3 = 0

Via DQUAD: 𝑥 = 8.4, 13.8

Homework Section 3.1

42.

a. 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 = 3𝑥 + 4𝑦 = 120

𝐴𝑟𝑒𝑎 = 𝑥𝑦

𝑦 = −34 𝑥 + 30

𝐴𝑟𝑒𝑎 = 𝑥(−34 𝑥 + 30)

𝐴𝑟𝑒𝑎 = −34 𝑥

! + 30𝑥

𝑉𝑒𝑟𝑡𝑒𝑥!!!""# = −𝑏2𝑎 =

−30

2 − 34=−30

− 32= 20

𝑦 = −34 20 + 30 = −15+ 30 = 15

The dimensions of an individual coop are 20𝑓𝑡 ×15𝑓𝑡.

b. The maximum area of an individual coop is 300𝑓𝑡!.

𝐴𝑟𝑒𝑎 = 𝑥𝑦 = 20 15 = 300