C. Trigonometric Functions · Sine Functions

Real-World Changes and Graph Transformations

C2.5About 70 minutes

Learning goals

I can describe how changing a real situation changes its periodic graph.
I can match changes in a Ferris wheel or swimming context to the parameters a, c and d.
I can explain when a change affects the period instead of the amplitude.
I can write a new model after the situation changes.

Words to know

model
An equation or graph that describes a real situation, usually approximately.

horizontal compression
A squeeze of a graph toward the y-axis, which shortens the period.

horizontal stretch
A stretch of a graph away from the y-axis, which lengthens the period.

What happens to the graph when the situation changes?

So far you have changed equations and watched graphs move. Now go the other way: start with a real situation, change something about it, and predict what happens to the graph. Does a bigger Ferris wheel make the graph taller, wider or higher? Does a faster swimmer change the amplitude or the period? Linking the story to the transformation is what lets you build and adjust real models.

Remember from Lesson 9

In y = a sin(x − d) + c: |a| is the amplitude, c is the axis, and d is the horizontal shift. In this unit the period of the sine function in degrees is always 360°; but when the x-axis shows time, a real situation can also change how long one cycle takes.

A swimmer doing lengths

A swimmer swims lengths of a 25 m pool at a steady 1 m/s, turning instantly at each end. Their distance from the starting end goes up from 0 m to 25 m in 25 s, back to 0 m at 50 s, and repeats. The graph is periodic (but not sinusoidal: it has sharp corners at the turns).

Period = 50 s (one length out and one length back).
Maximum = 25 m (the far end), minimum = 0 m.
Axis: d = 12.5, amplitude = 12.5 m.

Solid: 25 m pool at 1 m/s (period 50 s, max 25 m). Dashed: a 50 m pool at the same speed: the maximum doubles and so does the period. Dotted: a 25 m pool at 2 m/s: the same maximum, but the period halves to 25 s.

Worked example

Example 1: Changing the swim

For the 25 m swimmer, describe how the graph changes in each case. (a) The swimmer starts 10 s later. (b) The swimmer swims at 2 m/s. (c) The swimmer swims in a 50 m pool at 1 m/s. (d) Distance is measured from the far end instead of the start end.

(a) Every event happens 10 s later: a horizontal translation 10 s to the right. Max, min and period stay the same.
(b) Each length takes 25 ÷ 2 = 12.5 s, so the period becomes 25 s: a horizontal compression (the cycle is squeezed to half its width). Max and min do not change.
(c) The far end is now 50 m away: max = 50 m. A length takes 50 s, so the period is 100 s. The graph is stretched vertically and horizontally by 2.
(d) When the swimmer is 0 m from the start, they are 25 m from the far end, and vice versa. The new graph is the old one reflected in the x-axis and translated 25 m up (d_new = 25 − d_old). Its max, min, axis and period are unchanged, but the graph starts at its maximum instead of its minimum.

(a) right 10 s; (b) period halves (horizontal compression); (c) max and period both double; (d) reflection in the x-axis plus a translation up 25.

A Ferris wheel in degrees

For a Ferris wheel seat, let θ be the angle the wheel has turned since the seat boarded at the bottom. Then the height is

h(θ) = a sin(θ − 90°) + c

where a = the radius and c = the height of the centre. The shift of 90° is there because the seat starts at the bottom (a minimum), and y = sin(x − 90°) has its minimum at x = 0°.

Check at θ = 0°: h = a sin(−90°) + c = −a + c, which is the centre height minus the radius: the bottom ✓.

Each feature of a real Ferris wheel controls one feature of its height graph.

Worked example

Example 2: Changing the wheel

A Ferris wheel has radius 15 m and centre 17 m above the ground, so h(θ) = 15 sin(θ − 90°) + 17. Write the new equation and describe the change to the graph if (a) the wheel is rebuilt on a platform 5 m higher, (b) a new wheel has radius 18 m and centre 20 m, (c) you board a seat that is at the 3 o'clock position (level with the centre, on the side that is about to go up) when the wheel starts.

(a) Every height increases by 5: h = 15 sin(θ − 90°) + 22. A vertical translation 5 m up. Max 37 m, min 7 m.
(b) a = 18 and c = 20: h = 18 sin(θ − 90°) + 20. The amplitude grows from 15 to 18 (vertical stretch by 18 ÷ 15 = 1.2) and the axis rises from 17 to 20. Max 38 m, min 2 m (the bottom is still 2 m above the ground).
(c) The seat starts on the axis going up, exactly like y = sin x at 0°, so d = 0°: h = 15 sin θ + 17. The graph is translated 90° to the left compared with the original.

(a) h = 15 sin(θ − 90°) + 22, up 5; (b) h = 18 sin(θ − 90°) + 20, taller and higher; (c) h = 15 sin θ + 17, 90° left.

Solid: the original wheel, h = 15 sin(θ − 90°) + 17. Dashed: the same wheel on a 5 m platform (translated up 5). Dotted: the larger wheel, h = 18 sin(θ − 90°) + 20 (taller and higher; its minimum is still 2 m).

Investigating with technology

In a graphing app (degree mode), enter h = a sin(x − d) + c with sliders for a, d and c. Set a = 15, d = 90, c = 17. Then act out each story: "build a bigger wheel" (drag a), "raise the wheel" (drag c), "board a different seat" (drag d). Before each drag, predict what the graph will do.

Worked example

Example 3: Telling the story from the graphs

Two graphs show the height of a seat on a wheel against the angle turned. Graph B has the same maximum and minimum as graph A, but every point of B is 60° to the left of the matching point on A. What could explain this?

Same max and min: same radius and centre height, so the same wheel (or an identical one).
Shifted 60° left: B reaches each height 60° of turning earlier. So B's seat starts 60° further around the wheel than A's (it is already part-way up when the wheel starts).

B's seat is 60° ahead of A's seat on the same wheel.

Just for fun

A swimmer told her coach, "I'm a periodic function: I keep coming back to the wall."

The coach said, "Great! Now let's work on your period."

Hands-on activity

Walk the pool in your hallway

You need: a hallway or room at least 4 m long, two sticky notes, a phone timer or stopwatch, a helper, paper and a pencil.

Mark a start line and an end line 3 m apart with sticky notes.
Walk at a steady pace from start to end and back, three times, without stopping. Your helper calls out the time every 2 s, and notes whether you are near the start, middle or end. (Or film it and pause later.)
Sketch your distance from the start against time. Estimate the period and maximum.
Repeat at a faster pace, then repeat starting 5 s late, then move the end line to 2 m. Sketch each graph on the same axes with a different line style.
Next to each new graph, name the transformation of the first graph.

What to look for: faster pace gives a shorter period (horizontal compression); starting late shifts the graph right; a 2 m course lowers the maximum to 2 m (vertical compression) and also shortens the period at the same speed.

Your turn

A Ferris wheel's height model is h = 10 sin(θ − 90°) + 12. The whole wheel is moved onto a 3 m platform. Write the new model.

For the same wheel, a seat boards at the top instead of the bottom. Write the model and describe the transformation.

A swimmer in a 25 m pool slows from 1 m/s to 0.5 m/s. What happens to the period and the maximum of the distance graph?

A wheel with radius 12 m is replaced by one with radius 20 m, with the bottom still 2 m above the ground. Find the new c and the new equation (boarding at the bottom).

Which parameter of h = a sin(θ − d) + c changes when you board a different seat on the same wheel?

The water depth in a harbour follows a sinusoidal pattern. On a day of especially big tides, high tide is higher and low tide is lower, with the same average. Which parameter changes?

More practice: warm-up

Raising a whole Ferris wheel changes which parameter?

A bigger radius changes which parameter?

Starting 10 s later shifts the time graph which way?

Does a faster-turning wheel change the max height?

For h = 8 sin(θ − 90°) + 10, what is the radius of the wheel?

More practice: on your own

A wheel's model is h = 25 sin(θ − 90°) + 28. (a) How high is the bottom? (b) The ride operator lowers the boarding platform so the bottom is 1 m above the ground; write the new model if the wheel is rebuilt with the same radius.

A swimmer swims lengths of a 25 m pool at 1 m/s but rests 10 s at each end. Is the motion still periodic? What is the period?

Graph A: h = 6 sin(θ − 90°) + 8. Graph B: h = 6 sin(θ − 30°) + 8. Describe how B differs from A and explain in terms of the seats.

The axis of a tide graph is at 4 m and the amplitude is 3 m. A storm surge raises the water level by 0.5 m all day. How do the max, min, axis and amplitude change?

A wheel turns once every 40 s. A faster setting turns it once every 30 s. Describe the change to the graph of height against time and against angle turned.

Challenge

Design a wheel whose height model is h = a sin(θ − 90°) + c with the top at 52 m and the bottom at 2 m. Then find the height after a turn of 150°.

Two seats on the same wheel are directly opposite each other. If one has model h = 10 sin(θ − 90°) + 12, find the other's model in two different ways.

For the swimmer measured from the far end (Example 1d), write a rule relating the new distance D to the old distance d, and explain why it is a reflection and a translation.

Check your understanding

Quiz

A Ferris wheel is moved to a hilltop 40 m higher. Which part of h = a sin(θ − d) + c changes?

A swimmer swims faster in the same pool. On the distance-time graph, what changes?

A wheel has h = 12 sin(θ − 90°) + 14. What is the height of the bottom?

You board a seat that is at the top when the wheel starts. Compared with boarding at the bottom, the graph of height against angle is

A bigger wheel has a larger radius but the same bottom height. Which parameters change?

On a day of bigger tides, high tide is higher and low tide is lower, but the average level is unchanged. This is

For parents and teachers

Big idea: Connecting changes in a real situation that can be modelled by a periodic function to transformations of its graph (C2.5).

How to help: Use any wheel or round-trip motion. Ask "what if" questions: "What if the wheel were bigger? Higher? You got on a different seat? It turned faster?" and have your teen answer with a transformation word (stretch, translation up, translation left/right, compression).

Common difficulties:

Thinking a faster wheel makes the seat go higher. Speed changes the period in time, not the amplitude.
Forgetting that a bigger wheel with the same bottom also has a higher centre (c changes too).
Mixing up 'later' (right) and 'earlier' (left) shifts.