D. Trigonometric Functions · Sinusoidal Functions

What If? Predicting How Changes Affect a Sinusoidal Model

D3.4 D3.2About 70 minutes

Learning goals

I can predict which parameter of a sinusoidal model changes when a condition in the situation changes.
I can write the new model after a change.
I can work backwards: from a change in the equation, describe what changed in the situation.

Words to know

condition
A feature of the real situation, such as the size of a wheel, its speed, or where a person starts.

motion sensor
A device that measures the distance to an object many times per second; the data can be graphed against time.

sensitivity of a model
How much a model's graph changes when one condition changes.

What happens if…?

Engineers who design a Ferris wheel, scientists who study tides and students with a motion sensor all ask the same kind of question: if one thing about the situation changes, how does the graph change? Because each parameter of y = a cos(k(x − d)) + c has a clear meaning, you can often answer without collecting any new data.

If this condition changes…	…this parameter changes	…and the graph
size of the circle or wheel (radius)	a	taller or shorter waves
height of the axle, or how far the circle is from the sensor	c	moves up or down
speed of turning or walking	k (period = 360 ÷ k)	waves closer together or farther apart
starting time or starting position	d (or the sign of a, or sine vs cosine)	slides left or right, or flips

How the parameters of a circular-motion model are linked to the situation.

Walking in a circle in front of a motion sensor

A motion sensor is fixed to a wall. It measures how far you are from the wall, straight out along the floor. You walk at a steady speed around a circle of radius 1.5 m whose centre is 3 m from the wall, taking 8 s per lap, and you start at the point nearest the wall.

The distance swings between 3 − 1.5 = 1.5 m and 3 + 1.5 = 4.5 m: c = 3, |a| = 1.5.
The period is 8 s: k = 360 ÷ 8 = 45.
You start at the minimum distance: reflected cosine.

d(t) = −1.5 cos(45t) + 3

Top view (looking down from the ceiling) of the motion-sensor walk. Only the distance straight out from the wall is measured, so it swings between 1.5 m and 4.5 m.

Worked example

Example 1: Predicting changes for the circle walk

Starting from d(t) = −1.5 cos(45t) + 3, write the new model if (a) you walk twice as fast; (b) the circle's centre is moved to 4.5 m from the wall; (c) the circle has radius 1 m; (d) you start at the point farthest from the wall.

(a) Twice as fast: one lap takes 4 s, so k = 360 ÷ 4 = 90. d = −1.5 cos(90t) + 3. Only k changes.
(b) The centre is the middle of the motion: c = 4.5. d = −1.5 cos(45t) + 4.5. Only c changes.
(c) A smaller circle: |a| = 1. d = −cos(45t) + 3. Only a changes.
(d) Starting at the farthest point means starting at a maximum, so use a cosine with a > 0: d = 1.5 cos(45t) + 3 (a reflection of the original graph in its axis). The same graph is also d = −1.5 cos(45(t − 4)) + 3, a phase shift of half a period.

(a) k doubles, (b) c changes, (c) a changes, (d) the graph is reflected (or shifted by half a period).

Walking twice as fast (dashed) halves the period from 8 s to 4 s. The nearest and farthest distances do not change.

Moving the circle farther from the wall (dashed) raises the whole graph. A smaller circle (thin) gives a smaller amplitude around the same axis.

A surprising non-change

What if you walk the circle in the opposite direction, still starting at the nearest point? The graph does not change at all. Going clockwise or counterclockwise, after the same time you are the same distance around the circle from the start, so you are the same distance from the wall. In symbols, cos(−θ) = cos θ. (If you started at the side of the circle instead, the direction would matter: one way you move away from the wall first, the other way toward it.)

Worked example

Example 2: A new Ferris wheel

A fair replaces its Ferris wheel, h = −20 cos(6t) + 22 (radius 20 m, axle 22 m, 60 s per turn), with a larger one: radius 25 m, loaded from the same platform 2 m above the ground, one turn every 90 s. Write the new model and describe how the graph changes.

The bottom stays 2 m up, so the axle is 2 + 25 = 27 m: c = 27.
Radius 25: |a| = 25. Period 90 s: k = 360 ÷ 90 = 4.
Riders still board at the bottom: h = −25 cos(4t) + 27.
Changes to the graph: the maximum rises from 42 m to 52 m while the minimum stays 2 m (taller waves, higher axis), and each cycle is longer (90 s instead of 60 s).

h = −25 cos(4t) + 27

Worked example

Example 3: Working backwards

A model for the circle walk changes from d = −1.5 cos(45t) + 3 to each equation below. Describe what changed in the situation. (a) d = −1.5 cos(30t) + 3 (b) d = −2 cos(45t) + 3 (c) d = −1.5 cos(45(t − 2)) + 3

(a) k went from 45 to 30, so the period went from 8 s to 360 ÷ 30 = 12 s: the person is walking more slowly.
(b) |a| went from 1.5 to 2: the circle's radius is now 2 m (same centre).
(c) A phase shift of 2 s, which is a quarter of a lap: the person reaches the nearest point at t = 2 s instead of t = 0. They started a quarter lap earlier on the circle, at the side, moving toward the wall.

(a) slower walking (12 s per lap); (b) bigger circle; (c) a different starting point.

Just for fun

Walking around in circles is usually a sign that you are lost. In this lesson, it is a sign that you are collecting excellent sinusoidal data.

Hands-on activity

Walk the circle

You need: a chair, masking tape, a tape measure, a piece of string about 1 m long, paper and a partner. Use a room with a clear wall and space to walk.

Put the chair 2.5 m from the wall as the centre. Use the string (held at the chair) to mark a circle of radius 1 m on the floor with 8 pieces of tape, equally spaced. Number them 0 to 7 going counterclockwise, with 0 at the point nearest the wall.
Stand on each mark in order, two laps (0, 1, 2, …, 7, 0, 1, …). At each mark, your partner measures your distance from the wall, straight out.
Graph distance against mark number. Write a model using k = 360 ÷ 8 = 45 (per mark).
Predict the new graph and model if (a) the chair moves to 3 m from the wall, (b) the radius becomes 0.6 m, (c) you start at mark 4. Then test one prediction.

What to look for: with centre 2.5 m and radius 1 m, the distances are about 1.5, 1.8, 2.5, 3.2, 3.5, 3.2, 2.5, 1.8, … m, and the model is d = −cos(45n) + 2.5. Moving the chair changes c, the radius changes a, and starting at mark 4 flips the graph (d = cos(45n) + 2.5).

Your turn

The Ferris wheel h = −20 cos(6t) + 22 speeds up to one turn every 45 s. Write the new model.

The same wheel is moved onto a base so that its axle is 25 m above the ground (same radius and speed). Write the new model.

The circle walk d = −1.5 cos(45t) + 3 is changed to a circle of radius 1 m with the same centre and speed. Write the new model.

What change to the circle walk gives d = −1.5 cos(45t) + 5?

What change to the circle walk gives d = −1.5 cos(30t) + 3?

Compared with southern Ontario, a community much farther north has longer summer days and shorter winter days. Which parameter of the daylight model changes the most?

A weight bouncing on a spring is replaced with a heavier one, and it bounces more slowly. Which parameter changes, and in which direction?

More practice: warm-up

You walk around a bigger circle. Which parameter changes?

You walk faster. Does k get bigger or smaller?

The circle is moved farther from the sensor. Which parameter changes?

You start at the farthest point instead of the nearest. What happens to the graph?

The period changes from 8 s to 4 s. How does k change?

More practice: on your own

The Ferris wheel h = −20 cos(6t) + 22 is replaced with one of radius 15 m, still loaded from a platform 2 m above the ground, at the same speed. Write the new model.

On the original wheel (h = −20 cos(6t) + 22), a rider boards at the bottom at t = 10 s instead of t = 0. Write the model.

A motion-sensor graph for a circle walk has a minimum of 1 m at t = 0, a maximum of 3 m, and a period of 10 s. Describe the circle and the walk.

A southern Ontario daylight model is D = 3.2 sin(0.986(n − 80)) + 12.2. A northern community has a daylight amplitude of about 6 h. Assuming the same axis and phase shift, write its model and give its longest day length.

Explain why walking the circle in the opposite direction (starting at the nearest point) gives the same motion-sensor graph.

Challenge

Two people walk the same circle at the same speed (d = −1.5 cos(45t) + 3 for the first). The second starts at the farthest point. Write the second model and find the first three times when they are the same distance from the wall.

A walker starts at the side of the circle (3 m from the wall). Write models for walking first away from the wall and first toward the wall. Why does direction matter here?

A wheel's model is h = −18 cos(10t) + 20. The owner wants the period 50% longer and the top of the ride 5 m higher, with the bottom staying where it is. Write the new model.

Check your understanding

Quiz

A person walking in a circle in front of a motion sensor starts walking faster. Which parameter changes?

The person walks a larger circle with the same centre. Which parameter changes?

A Ferris wheel's axle is raised 3 m and nothing else changes. What happens to h = −20 cos(6t) + 22?

A Ferris wheel model changes from h = −20 cos(6t) + 22 to h = −20 cos(3t) + 22. What happened?

A motion-sensor walker starts at the point farthest from the sensor instead of the nearest. What happens to the model d = −1.5 cos(45t) + 3?

Compared with southern Ontario, how does a daylight model for a community farther north change?

For parents and teachers

Big idea: D3.4 asks students to predict, by interpreting the parameters, how a model changes when the conditions of the situation change (for example, walking in a circle in front of a motion sensor). Students should connect each physical change to one parameter.

How to help: The "Walk the circle" activity is quick and memorable; help with the measuring. Ask "What if…?" questions: "What if the wheel were twice as big?" "What if you started at the other side?"

Common difficulties:

Changing a and forgetting that c must change too when the bottom of a Ferris wheel stays at the platform.
Thinking "faster" means a bigger period. Faster means a shorter period and a bigger k.
Assuming every change affects the graph. Reversing direction from the nearest point does not.
Not explaining in context. A full answer names the condition, the parameter, and how the graph looks different.