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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