Math 3 Unit 6, Lesson 5, Applying Trig Graphs
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𝜋
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1) What are the period and amplitude of 𝑦 = − 𝑠𝑖𝑛 ( 𝑥) + 1 ?
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2) Graph using transformations: 𝑦 = 2𝑐𝑜𝑠 ( 𝑥).
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3) A car's tire has a diameter of 32 inches. It runs over a nail, but it
can continue moving. Write a cosine function that describes the
height of the nail above ground as a function of angular distance.
4) A Ferris wheel is 4 feet off the ground. It has
a diameter of 26 feet, and rotates once every 32 seconds. If
you begin the ride sitting in a chair that is 17 feet above the
ground, how high will you be 10 seconds into the ride? During
the first minute, when will you be 20 feet high?
5) Each day, the tide continuously goes in and out, raising and lowering a boat
(sinuisoidally) in the harbor. At low tide, the boat is only 2 feet above the
ocean floor. And, 6 hours later, at peak high tide, the boat is 40 feet above
the ocean floor. Write a sine function that describes the boat's distance above
the ocean floor as it relates to time. For safety, the boat needs 25 feet of
depth to sail. If high tide occurs at 3 PM, between what day light times can
the boat go out to sea?
6) The following trig function models the position of a rung on a waterwheel:
𝜋
𝑦 = −20𝑠𝑖𝑛 ( 𝑡) + 16
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where t = seconds and y = number of feet above water level.
a) What is the diameter of the wheel?
b) At the top of the wheel, how high is the rung above water level?
c) How many rotations per minute does the wheel make?
Math 3 Unit 6, Lesson 5, Applying Trig Graphs
7) The motion of a swing hanging from a tree next to a lake can be modeled by
a sinusoid. The tree is 10 feet from the water, and the swing
can extend 20 feet from the tree in each direction. If it takes
2 seconds to swing from one side to the other side, write
cosine function that models the position of the swing as a
function of time and determine the interval of time that the
swing is above the water. Assume the swing is pulled back
before releasing.
8) The London Eye is a huge Ferris wheel with diameter 135 meters (443 feet)
in London, England, which completes one rotation every 30 minutes. Riders
board from a platform 2 meters above the ground. Express a rider’s height as
a function of time.
9) The hours of daylight in Seattle oscillate from a low of 8.5 hours in January
to a high of 16 hours in July. When should you plant a garden if you want to
do it during the month where there are 14 hours of daylight?
10)
Use SinReg to model the home owner’s monthly gas usage (in
therms). Round to four decimals in your model.
Math 3 Unit 6, Lesson 5, Applying Trig Graphs
Answer Key
1) Amplitude = 4/3. Period = 4
2)
3) 𝑦 = −16𝑐𝑜𝑠(𝛳)+16
𝜋
4) 𝑓(𝑥) = 13𝑠𝑖𝑛 ( 𝑥) + 17, 𝑓(10) = ~29 𝑓𝑒𝑒𝑡, 20 𝑓𝑒𝑒𝑡 =
16
𝑓(~1.19 seconds) = 𝑓(~14.81sec) = 𝑓(~33.18 sec) =
𝑓(~46.81 sec)
𝜋
5) 𝑓(𝑥) = 19𝑠𝑖𝑛 ( 𝑥) + 21, 25 𝑓𝑒𝑒𝑡 = 𝑓(12.40) = 𝑓(17.59).
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12.40 = 12: 24 𝑃𝑀
17.59 = 5: 35 𝑃𝑀
2π
6) Diameter = 40 feet. Maximum = 36 feet. 𝛵 = 𝜋 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 =
12
⁄6
𝑠𝑒𝑐𝑠
𝑟𝑒𝑣𝑜𝑙𝑢𝑡𝑖𝑜𝑛
→ 5 𝑟𝑒𝑣/𝑚𝑖𝑛
𝜋
7) 𝑦 = 20𝑐𝑜𝑠 ( 𝑡). It is over the water from 1.33 seconds to 2.67 seconds.
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𝜋
8) ℎ(𝑡) = −67.5𝑐𝑜𝑠 ( 𝑡) + 69.5
15
𝜋
9) ℎ(𝑡) = −3.75𝑐𝑜𝑠 ( 𝑡) + 12.25. ℎ(𝑡) ≥ 14 for
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[3.93, 8.07] or about March 29th to August 2nd.
10)
𝑦 = 64.8664𝑠𝑖𝑛(0.5257𝑥 + 0.5081) + 87.7352