IB Math SL Year 2
Name: __________________________________
Date: _______________________
5-10: Modeling with Trig Graphs
In this lesson, we will use trigonometric functions to model real-life situations that show periodic behavior.
Let’s see how this works…
1. The height of water in a harbor is 16m at high tide and 10m at low tide,
which occurs 12 hours later. The graph below shows how the height of
water changes with time over 24 hours.
(a) Find an equation for the height of water (in metres) in terms of time (in
hours) in the form h = m + acos(bt)
(b) Find the first two times after high tide when the height of water is 12m.
Steps
My Work
PART A – writing an equation
Prep
1. What variables/attributes do
I need for a complete
equation?
2. Amplitude
3. Frequency
(We get this from the period)
4. Vertical Shift
(We get this from the midline)
5. Putting it all together
6. Use equation to substitute
given information.
(Use GDC to find intersection
points.)
PART B- Analyzing a given situation
IB Math SL Year 2
Try again…
2. A Ferris wheel with diameter 122 meters rotates clockwise at a
constant speed. The wheel completes 2.4 rotations every hour.
The bottom of the wheel is 13 meters above the ground.
A seat starts at the bottom of the wheel.
a) Find the maximum height above the ground of the seat.
After t minutes, the height, h meters above the ground of the seat is given by: ℎ = 74 + 𝑎 cos 𝑏𝑡
b) Show that the period of h is 25 minutes.
c) Using part b) write down the exact value of b.
d) Find the value of a.
e) Write the equation, ℎ = 74 + 𝑎 cos 𝑏𝑡.
Think!! How do we know if a is positive or
negative?
IB Math SL Year 2
f) Sketch the graph of h for, 0 ≤ 𝑡 ≤ 50
g) In one rotation of the wheel, find the amount of times the first seat given seat is 105 meters above the ground.
3. NO CALCULATOR
The following diagram shows a waterwheel with a bucket. The wheel
rotates at a constant rate in a counterclockwise direction.
The diameter of the wheel is 8 metres. The centre of the wheel, A, is
2 metres above the water level. After t seconds, the height of the
bucket above the water level is given by h = asin(bt)+2.
(a) Show that a = 4.
The wheel turns at a rate of one rotation every 30 seconds.
𝜋
(b) Show that b = 15
IB Math SL Year 2
𝜋
4. The depth of water in a harbor varies during the day and is given by the equation d = 16 + 7sin(12t), where d is
measured in metres and t is the number of hours after midnight.
a. Find the depth of the water at low and high tide.
b. At what times does the high tide occur?
5. The graph shows the depth of water below a walkway as a function of time. The equation of the graph is of the form
y = acos(bt) + m. Find the values of a, b, and m.
IB Math SL Year 2
6. The height¸ h metres¸ of a seat on a Ferris wheel after t minutes is given by h(t) = -15cos 1.2t + 17, for t > 0.
a. Find the height of the seat when t = 0.
b. The seat first reaches a height of 20m after k minutes. Find k.
c. Calculate the time needed for the seat to complete a full rotation, giving your answer correct to one decimal
place.
7. Let f(x) = acos(b(x-c)). The diagram below shows part of the graph of f, for 0 < x < 10.
The graph has a local maximum at P(3,5), a local minimum at Q(7,-5), and crosses the x-axis at R.
(a) Write down the value of
i.
a
ii.
c
(b) Find the value of b.
(c) Find the x-coordinate of R.