1
Answer Key for RSG’s Secondary 3 Honors 2.6H_RSG b) GO READY x 𝑔 𝑥 = 4 !
12.
7! = 1 1 a.
linear -‐2 𝟏 13.
𝑎 ! = 1 𝟏𝟔
b.
𝑓 −3 = −23; 𝟏 14.
6! = 6 -‐1 𝑓 𝑥 = 𝑓 𝑥 − 1 + 6 𝟒 15.
𝑙𝑜𝑔! 𝑎 = 1 because 𝑎! = 𝑎
1 16.
𝑙𝑜𝑔! 4 ! = 𝑥 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 4 ! = 2 a.
quadratic
b.
𝑓 −3 = 4; 1 4 4 !
𝑓 𝑥 = 𝑓 𝑥 − 1 + 2𝑥 2 16 17.
𝑙𝑜𝑔! 𝑎 ! = 𝑥 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑎 ! = 3 a.
linear 𝑎 !
b.
𝑓 −3 = −15; c) 18.
x = y 𝑓 𝑥 = 𝑓 𝑥 − 1 + 5 x ℎ 𝑥 = 𝑒 !
19.
= 81 because 4 a.
linear -‐2 𝟏 ≈ 𝟎. 𝟏𝟑𝟓𝟑 𝑙𝑜𝑔! 81 equals the exponent u 𝒆𝟐
b.
𝑓 −3 = 24; 𝟏 ≈ 𝟎. 𝟑𝟔𝟕𝟗 that make 3! = 81 -‐1 𝒆 𝑓 𝑥 = 𝑓 𝑥 − 1 − 2 20.
= 𝑥 because 𝑙𝑜𝑔! 𝑥 equals
1 the exponent u that make 5 a.
quadratic
b.
𝑓 −3 = 48; 1 𝒆 ≈ 𝟐. 𝟕𝟏𝟖𝟑 𝑎 ! = 𝑥 𝑓 𝑥 = 𝑓 𝑥 − 1 + 10𝑥 − 6 2 𝒆𝟐 ≈ 𝟕. 𝟑𝟖𝟗𝟏 6 a.
quadratic 2.7H_RSG
b.
𝑓 −3 = 4; graph READY 𝑓 𝑥 = 𝑓 𝑥 − 1 + 2𝑥 + 1 𝑓 𝑥 = 2 ! 𝑏𝑙𝑢𝑒 1.
𝑔 𝑥 = 4 ! 𝑟𝑒𝑑 x 𝑦 = 𝑥 !
SET 7 a.
$16,776 ℎ 𝑥 = 𝑒 ! 𝑔𝑟𝑒𝑒𝑛 -‐3 -‐3
-‐2 -‐2
b.
$16,991
-‐1 -‐1
c.
$ 17,103 0 0
d.
$17,216 1 1
e.
$17,217.22
2 2 8 a.
$801,423
3 3
b.
$893,815
c.
$946,916
2.
d.
$1,004,799.01 x 𝑦 = 𝑥 !
e.
$1,005,472.99 -‐3 9
-‐2 4
9 a) -‐1 1 x 𝑓 𝑥 = 2 !
0 -‐2 𝟏 10.
Point (0,1) ; Any number 𝟒 raised to the zero power 1 1 -‐1 𝟏 2 4 𝟐 equals one.
1 3 9 1 2 11.
a) 𝑓 𝑥 < ℎ 𝑥 < 𝑔 𝑥
2 4 b) 𝑔 𝑥 < ℎ 𝑥 < 𝑓 𝑥
2
2.7H_RSG continued 6.
The table and the graph 9.
10000: 2.5 and 2.75 hours 3.
both indicate two points in 1000000: 4.25 and 4.5 hours x 𝑦 = 𝑥 !
common (0,0) and (1,1).
10.
5.5 X 1029 Logical because 0 raised to any !”# !”” 11.
𝑡 = !
-‐3 -‐27 positive power is still 0 and 1 ! 12.
1.661 hours -‐2 -‐8 raised to any power is 1.
-‐1 -‐1 GO 7.
The graphs of even powers
0 13.
-‐0.2 14.
-‐0.8 are in quadrants 1 & 2 or 1 1 15.
-‐0.1 16.
-‐0.6 always positive.
The graphs of 17.
3.18 18.
3B 2 8 odd powers are in quad.
1 & 3.
19.
-‐0.2 20.
A + C – B 3 27 Even reflect across y-‐axis Odd reflect about the origin, if _____________________________ 4.
the function has NOT been x 𝑦 = 𝑥 ! translated) 2.8H_RSG
READY -‐3 81 SET 1.
100 2.
7 -‐2 16 3.
27 4.
4 8.
-‐1 1 time (hours) #bacteria 5.
7 6.
131
0 0 7 7.
71 8.
3
1 1 1 112 SET 2 16 2 1792 9.
b 10.
a 11.
c 3 81 3 28672 12.
a 13.
b 14.
a 5.
4 458752 15.
b 16.
c 17.
b 5 7340032 18.
𝑙𝑜𝑔! 50 = 1 + 𝑙𝑜𝑔! 2 !” Equation: 𝑦 = 7 ∗ 2 !!
19.
𝑙𝑜𝑔! = 𝑙𝑜𝑔! 32 − 𝑙𝑜𝑔! 4 !
Graph: 20.
𝑙𝑜𝑔 90 = 𝑙𝑜𝑔3 + ! x4 x-‐scale 1 hour; ! 21.
all 3 are equal x2 y-‐scale 5000 bacteria
GO
22.
5; 23.
-‐3; 24.
-‐1
25.
-‐7; 26.
19; 27.
4
!”
28.
-‐3; 29.
-‐2; 30.
!
2.9H_RSG x READY
1.
-‐2, 1
2.
3
3.
1
4.
0, 2, -‐2 x3 5.
𝑎 ) 𝑦 = 𝑙𝑜𝑔! 𝑥; Reasons may
vary.
The curve looks
logarithmic.
The 3 points (1,0),
(2,1), (4,2) suggest base 2.
3
2.9H_RSG continued 26.
2 + 𝑙𝑜𝑔! 𝑤 − 𝑙𝑜𝑔! 𝑥 − !”# ! !”# ! 16.
∙ = 𝑠𝑖𝑛𝜃 !”# ! ! 𝑏) 𝑦 = 𝑥 ! + 𝑥 − 2; Reasons 𝑙𝑜𝑔! 𝑦 − 𝑙𝑜𝑔! 𝑧 17.
1 + cos 𝛽 1 − cos 𝛽 = may vary.
Since x = -‐2 and 1, 27.
a) 4 1 − 𝑐𝑜𝑠 𝛽 ! = 𝑠𝑖𝑛! 𝛽 = the factors (x+2)(x-‐1) will b) 7 1 = 𝑠𝑖𝑛! 𝛽 + 𝑐𝑜𝑠 𝛽 ! ; 𝑝𝑦𝑡ℎ𝑎𝑔 𝐼𝐷 generate a possible equation.
c) 8 18.
Do the same as 17.
The y-‐intercept -‐2 confirms d) 9 19.
Replace 𝑐𝑜𝑠 ! 𝑊 with the exact equation.
e) 9 1 + 𝑠𝑖𝑛! 𝑊; 𝑠𝑖𝑚𝑝𝑙𝑖𝑓𝑦
𝑐 ) 𝑦 = 𝑥 ! − 4𝑥; Reasons f) 10 20.
Multiply to get may vary.
The x-‐intercepts and 𝑐𝑜𝑠 ! 𝑥 − 𝑠𝑖𝑛! 𝑥 the max and min confirm the
𝑑𝑜𝑢𝑏𝑙𝑒 𝑎𝑛𝑔𝑙𝑒 𝐼𝐷 equation.
21.
Add to get 2 sin 𝑢 𝑐𝑜𝑠 𝑢 𝑑) 𝑦 = 2𝑥 − 6; Reasons may
𝑑𝑜𝑢𝑏𝑙𝑒 𝑎𝑛𝑔𝑙𝑒 𝐼𝐷 vary.
Slope = 2, Point (3,0) 6.
Answers may vary.
The x-‐
GO intercepts can be used to 4.7H_RSG ! !! 22.
30°, 150°, , generate factors for the ! ! !! !!! equations.
They also indicate 23.
210°, 330°, , ! ! the degree of the function.
6.14H_RSG 24.
45°, 315°, , ! !!
SET READY ! ! !! !! 7.
4 8.
5, -‐5 Graphs for 1 (dash/dot; blue) 25.
240°, 300°, , ! ! 9.
4, -‐5 10.
8 and 3 (dotted; red) !! !! 26.
135°, 315°, ; ! ! 11.
4 12.
3, 4 ! !! 27.
60°, 240°, , 13.
1 4 14.
12 5 ! !
15.
3 1 2 𝑜𝑟 7 2 6.15H_RSG !”# ! !! 16.
𝑜𝑟 ≈ 0.9515 READY ! 17.
10 18.
2 7 1.
10 units 2.
61 𝑢𝑛𝑖𝑡𝑠 19.
a.
50,118,723.36 3.
7 2 𝑢𝑛𝑖𝑡𝑠 4.
5 units
b.
7,943,282.35 5.
10 𝑢𝑛𝑖𝑡𝑠 6.
4 units
c.
!”##$%&’.!” ≈ 6.3 SET !”#$%&%.!” 7.
0.175 radians GO 8.
1.693 radians 20.
4,3,5,2,1 21.
4,2,1,3,5 9.
4.8 radians
10.
3.508 radians 22.
a) 𝑙𝑜𝑔! 729 = 6 2.
𝐺! = 𝑓 𝑡 + 5 11.
0 radians
b) 𝑙𝑜𝑔! 0.04 = −2 4.
𝐺! = 𝑓 𝑡 − 2 12.
3.334 radians !
c) 𝑙𝑜𝑔 ! = −1 SET 13.
2.706 radians ! ! 23.
a) 6! = 216 5.
20 6.
62 7.
36 14.
5.847 radians
b) 9! = 1 8.
81 9.
18 10.
45 15.
2.356 radians !! ! !
c) 2!! = 0.5 11.
12.
13.
16.
4.712 radians ! !” ! 24.
1 + 3𝑙𝑜𝑔! 𝑥 ! ! ! ! 17.
Both sine and cosine equal ! 14.
a) b) = 25.
𝑙𝑜𝑔! 𝑚 − 𝑙𝑜𝑔! 𝑛 !” ! ! 1 and -‐1 in one location.
! ! ! ! 15.
a) b) c)
!” !” !”
4
6.15H_RSG continued 8.
13.
GO 18.
7.85 in 19.
10.47 cm 20.
28.27 ft 21.
3.08 mm
6.16_RSG 14.
READY 1.
0.3143 radians 9.
2.
0.6429 radians 3.
1.697radians 4.
2 radians SET 5.
GO
15.
141.37 m2
10.
6.
16.
1116.10 m2
11.
7.
12.
17.
2010.62 mm2
5
6.16H_RSG Part 2 GO 9.
Motion: Lines from (-‐5, 5)
25.
16 + 5𝑖 to (5, 5) to (-‐5, -‐5) to (5, -‐5) to READY 26.
7 − 11𝑖 (-‐5, 5) 1.
(-‐3, 4) 27.
−6 + 9𝑖 28.
−16 − 2𝑖 2.
(5, 2) 29.
6 + 2𝑖 3.
(1, -‐6) 30.
61 4.
red 5.
green 6.
blue 7.
brown
7.7H_RSG
10.
table
READY
1.
𝑦 = 4𝑥 − 11
2.
𝑦 = −5𝑥 − 7
! !” 3.
𝑦 = 𝑥 −
! !
! !
4.
𝑦 = − 𝑥 +
! !
! ! 5.
𝑦 = 𝑥 − graph:
! !
! 6.
𝑦 = 𝑥 + !
! ! 8.
1) 5 2) 29 3) 37
4) 34 5) 2 5 6) 37 SET
7) 5 7.
Motion: Lines from (0, 5) to equation:
(5, 0) to (-‐5, 0) to (0, -‐5)
𝑥 − 3 ! + 𝑦 − 2 ! = 1 SET
9.
(0, 4) GO ! ! 10.
, 11.
! ! 11.
(-‐2, 0) 12.
(5, -‐5) 13.
3 2, !!
!
14.
(-‐6, π) 8.
Motion: Lines from (-‐2.5, 5) 12.
15.
2, !
to (5, 5) to (-‐2.5, -‐5) to (-‐5, -‐5) !
!! to (-‐2.5, 5) 16.
8, !
13.
17.
10 cos 3.46 + 𝑖 𝑠𝑖𝑛 3.46 !! !!
18.
3 2 cos + 𝑖 sin ! ! 19.
65 cos 2.62 + 𝑖 𝑠𝑖𝑛 2.62 20.
2 cos + 𝑖 sin ! !
! ! 14.
! ! ! 21.
− + 𝑖 ! ! ! ! ! !
22.
− + 𝑖 ! !
23.
2 − 2𝑖
24.
8𝑖
6
7.8H_RSG Rectangular equation: !!!!! 12.
a)
!!!! 𝑦 = 𝑥 − 2 !
READY Graph: !”!!! 1.
mean 85.75, median 85, b) !!! mode 85 !”! 2.
Many possible answers; c) !” Example: Only one of the y-‐ values is actually 4.
All of the !”
d) = 17 rest are either more than 4 or ! less than 4.
If the distance of each point above the y = 4 line Motion: Beginning at (2, 0) 13.
Yes; ! is considered positive and the moves up to the right in a a) 𝑥 ≠ 1, − ! distance of each point below parabolic curve b) 𝑥 ≠ 1 the line is considered negative, 5.
Table: those distances will cancel each The numbers 10 and 4 are not other out.
values that would make the
functions undefined, so there SET are no restrictions on c) or d).
3.
Table:
Rectangular Equation:
𝑥 − 5 ! + 𝑦 − 3 ! = 4
Graph:
Rectangular equation: 2 𝑦 = 𝑥 + 1 3 Graph:
Motion: Moves counter-‐ clockwise beginning at (7, 3)
GO Motion: Beginning at (-‐3, 1) 10.
a) 14 − 3𝑥 !
moves up to the right in a line b) −9𝑥 ! − 12𝑥 4.
Table: c) −34
d) −12
11.
No; there are no values of x that make the equations undefined.
