Chris Kirkpatrick, Barbara Alldred, Andrew Dmytriw, Shawn Godin, Angelo Lillo, David Pilmer, Susanne Trew, Noel Walker · Grade 11 · Study material generated by AI from this edition. Not affiliated with the publisher.
Chapter 6 · pp. 352–421
This chapter studies graphs that repeat, such as daylight hours, a spinning wheel, or a swaying flagpole. It begins with periodic functions and their features (period, axis, amplitude, range). It then narrows to sinusoidal functions, the smooth waves made by sine and cosine, and shows how the numbers in transform the graph. Finally, it uses these functions to model real situations.
A periodic function has a graph that repeats the same pattern at equal intervals. In a table of values, the same y-values come back each time x goes up by a fixed amount. Examples are the hours of daylight at one place over the years, and the height of a point on a spinning cylinder (p. 357).
One repeating piece of the graph is a cycle. The period is the change in x needed to complete one cycle. You can measure it between any two matching points, such as peak to peak or trough to trough (pp. 357, 359).
Example (daylight at Hudson Bay). The longest days are at day 166 and day 531, so the period is days. The shortest days, at days 349 and 714, are also 365 days apart. This agrees with the period (pp. 356, 357).
A peak is a maximum point on the graph, and a trough is a minimum point (p. 359).
Example: a metal strip is spooled onto a rotating cylinder, and the height of its end is graphed against time. The troughs are at and s, so the period is 0.25 s. Measuring from peak to peak gives the same answer (p. 359).
The axis is the horizontal line halfway between the maximum and the minimum:
The strip end moves between 1 mm and 9 mm, so the axis is . For the daylight graph with a maximum of 18 h and a minimum of 6 h, the axis is (pp. 360, 361).
The amplitude is half the difference between the maximum and the minimum:
It is also the distance from the axis up to a peak or down to a trough. For the strip end, mm, and mm gives the same value. Amplitude is a distance, so it is never negative (p. 360).
The range of a periodic function that has a maximum and a minimum runs from the minimum to the maximum. The strip end stays between 1 mm and 9 mm, so the range is (pp. 359, 361).
To test a graph, ask whether the same shape repeats over the same interval each time. If it does, you can state the period, axis and amplitude (p. 361).
To predict a value in a periodic situation, follow these steps (pp. 356, 361):
Going beyond the given data in this way is called extrapolation. For Hudson Bay, the data cover about two years. Extending the 365-day pattern to day 943 (August 1 of year 3) gives about 17 h of daylight (p. 356).
When the height of a turning object is graphed against time, one period is the time for one complete revolution. The strip-end graph has a period of 0.25 s, so the cylinder makes one full turn every 0.25 s. The same idea works for Ferris wheels, saw blades and bicycle tires (pp. 359, 369).
The shape of the graph also tells you how the object moves (pp. 357, 358):
Example (cutting blade). The graph repeats every 4 s, so the period is 4 s. The blade's height is at most 0.5 cm, and a height of 0 cm means it is hitting the cutting surface. The graph is flat from 0 to 3.0 s and again from 4.0 to 7.0 s, so the blade rests for 3 s while the next 6 m length of strip is pulled through. It then takes 0.5 s to go down (3.0 s to 3.5 s) and 0.5 s to go back up (3.5 s to 4.0 s). The blade hits the surface at 3.5 s, 7.5 s and 11.5 s. After that it hits every 4 s, which is how you predict the next hit (p. 358).
A sinusoidal function is a periodic function whose graph is a smooth, symmetrical wave. Any piece of the wave can be slid horizontally onto another piece. Its graph can be built by transforming or . For example, makes a string of identical smooth waves, so it is sinusoidal (pp. 369, 374).
Every sinusoidal function is periodic, but not every periodic function is sinusoidal. Some graphs repeat but have lopsided waves. Others repeat with sharp corners, like the motion of a blade that stops and starts. These are periodic only (pp. 369–370).
To classify a graph, ask two questions in order:
Think of a water wheel of radius 1 m turning counterclockwise. A nail starts at water level, which is the point (1, 0), and the wheel turns through an angle . The nail's height above the horizontal line is . Its horizontal distance from the vertical line through the centre is . At the nail is at the top, so and (pp. 357–359, 372).
A bigger circle works the same way, scaled by the radius. A point starting at and rotated by about the origin lands at
This follows from similar triangles. In the radius- triangle, and . Rotating by gives (pp. 371–373).
Sine. The period is . The maximum is 1, the minimum is , and the amplitude is 1. The axis is . The domain is all real numbers, and the range is . The zeros are at (p. 372).
Cosine. It has the same period, amplitude, axis and range. It starts at its maximum, and its zeros are at (p. 373).
The two curves are congruent. Translating the sine curve to the left gives the cosine curve. Translating the cosine curve to the right gives the sine curve. This means either function can model the same wave with a different starting point (pp. 373, 375).
Five key points, spaced a quarter period () apart, sketch one cycle quickly (pp. 369, 372–373):
Plot the points and join them with a smooth curve.
To evaluate at , put the calculator in degree mode. Then work in this order (p. 371):
Radian mode would give a wrong answer.
To find the period from a graph drawn on a window from to , count the complete cycles and divide by that count. The graph of has 3 cycles, so the period is . Its maximum is 6 and its minimum is . The axis is , the amplitude is , and the range is (pp. 370–371).
Common wrong belief: "Any function with sine or cosine in its equation is sinusoidal." This is false. The functions and have waves that drift or grow, so they never repeat exactly. Only a transformed sine or cosine of the form is sinusoidal (p. 373).
A pebble in a tire rises and falls as the wheel turns, so its height against time is sinusoidal. One period is the time for one full revolution. Suppose Bike A's graph runs between 0 and 60 cm and Bike B's runs between 0 and 50 cm.
The speed of a point on a rotating circle is the circumference divided by the period:
Bike A: cm/s. Bike B: cm/s. Both bikes travel at about 3.14 m/s (pp. 366, 369).
So wheels of different sizes can go at the same speed if the larger wheel has a proportionally longer period. A faster bike with the same wheel would have a shorter period, and its graph would be compressed horizontally (pp. 365–367).
The same ideas apply to the height of one tooth on a circular saw blade, measured from the cutting surface. The amplitude equals the blade's radius. The axis shows how far the axle is above or below the surface (pp. 377–378).
The period is easiest to find by measuring from one peak to the next. Here that is 0 s to 0.02 s for Saw A and 0 s to 0.03 s for Saw B (p. 378).
Sine and cosine functions can describe anything that repeats in smooth, symmetrical waves at regular intervals. Examples include objects in circular orbits, a swinging pendulum, the hours of daylight through the year, the phases of the Moon, and the current in an AC circuit. The size of the wave depends on the situation. For a wheel, the amplitude is the radius. For a swing, it is how far the swing travels from its resting position. One full cycle of the motion is one period of the function. (pp. 379, 384–385)
The sine part finishes one cycle when the angle has grown by . So the period is
Example: has period . Its amplitude is 4.1 and its axis is . (pp. 379, 388)
To answer questions from a sinusoidal model:
In the employment model, is in years since May 1992 and is in hundreds of applications per week. The peaks are at and , so the cycle is years. This matches . The lowest value is , which means 170 applications per week. For , the angle is and . That is about 188 applications in May 2002. (p. 379)
Places farther from the equator have a bigger change in daylight over the year. Every location has a period of one year, or 365 days. Only the amplitude differs.
So a higher latitude gives a graph with a greater amplitude. (pp. 381–382)
Any sinusoidal function can be written as
Each letter transforms the parent graph or in its own way. Because the effects are independent, you can apply them one at a time. (pp. 387–388)
The value : amplitude and reflection. The amplitude is . Compared with :
The zeros and the period do not change. (p. 387)
The value : vertical translation. In , every point moves up units (or down if is negative). The axis becomes . For , the axis is , the maximum is 3 and the minimum is 1. The amplitude and period stay the same, but the zeros move, and there may be none. (p. 387)
The value : period. In the period becomes . The amplitude and axis stay the same.
(p. 388)
The value : horizontal translation. In , the graph slides units to the right when is positive. For example, shifts right. The period, amplitude and axis do not change. Only the positions of the maxima, minima and zeros move. (p. 388)
A common mistake is to think shifts right because of the plus sign. Rewrite it as . Then , so the shift is to the left. The direction of the shift is the opposite of the sign written inside the brackets. Compare , which moves right. (p. 388)
Example: has amplitude 3, period and axis . The pattern is shifted right. (p. 388)
The wave rises above the axis and falls below it. So for :
For , the maximum is , the minimum is , and the range is . The axis is . (pp. 386–387)
Some students say has amplitude . This is wrong. Amplitude is a distance from the axis to a maximum or minimum, so it is always positive. Here the amplitude is 2. The negative sign only tells you the graph is reflected in the x-axis. (pp. 384, 387)
A sinusoidal function is a sine or cosine curve that has been transformed. It is written or . The basic curves and repeat every . The four constants , , and only stretch, compress, reflect or slide that basic curve. For example, is a cosine wave that is taller and narrower than the basic one. (p. 390)
Each constant has its own job:
A reflection in the x-axis and a horizontal translation are the two transformations that leave the period, amplitude and axis unchanged. A reflection only flips the curve about the axis. A vertical translation changes the axis, a vertical stretch changes the amplitude, and a horizontal stretch changes the period. (p. 390)
For , and likewise for the cosine form:
For the amplitude is 5, the period is and the axis is . (p. 393)
A negative reflects the graph in the y-axis, but the period still uses . For the period is . (pp. 390, 396)
Range. Start at the axis and go one amplitude up and one amplitude down, so . With axis and amplitude 3, the range is . For the range is . The reflection does not change it. (pp. 393, 413)
Domain. The curve continues forever in both directions, so the domain is all real numbers. A real situation can restrict it. A time model has . A model that runs for three cycles has . (pp. 397, 400, 413)
Translations follow the signs of and in the standard form:
The bracket sign is the opposite of the direction. In we have , so the graph moves left . In the graph moves down 6. (p. 393)
The stretch factors are different for horizontal and vertical changes:
(p. 393)
The phase shift is the horizontal translation, which is . A positive is a shift right and a negative is a shift left. For the phase shift is 0 because there is no horizontal translation. (p. 391)
You do not need to move every point. Transform the five key points of one cycle of the basic curve, then join them with a smooth curve.
For the key point becomes . The x-coordinate is divided by 3 and the y-coordinate is multiplied by 2. (p. 392)
One step at a time. Start with and follow the order of operations: multiplication and division first, then addition and subtraction. The horizontal and vertical steps can be done in either order. (pp. 390, 392)
For :
(pp. 390, 392)
This function has amplitude 3, period , axis and a phase shift of right. Its minimum is and its maximum is , so the range is . (pp. 390–391)
Faster method. Apply all stretches, compressions and reflections together, then apply all translations. For :
The range is and the first minimum is at . This graph begins at a minimum, not a maximum, because is negative and there is no horizontal shift. (pp. 391–392)
A common mistake is to read the shift straight from the bracket. In it is tempting to say the graph moves left. The shift is only visible after you factor out of the bracket:
Now , so the graph moves to the left. The period is , and the graph is reflected in the x-axis. (p. 392)
When a graph is sinusoidal, you can rebuild its equation one feature at a time. You match each feature to one parameter in or . (pp. 397, 401, 413)
(p. 397)
Worked example: the water wheel. A nail on a water wheel has a height graph with a maximum of 3 and a minimum of . The peaks are 10 s apart.
(p. 397)
The domain is , because is elapsed time. (p. 397)
Cosine model. The basic cosine curve has a maximum at . If your graph starts at a maximum on the y-axis, no shift is needed, so . For the water wheel this gives . In general, for a cosine model is the x-coordinate of any maximum. If the nearest maximum is at , use . This is why a cosine model is usually easier when the graph begins at a peak. (pp. 397, 400, 413)
Sine model. The basic sine curve rises through its axis at . So must be the x-coordinate where your curve crosses the axis going upward. That point is a quarter period before the first maximum. For the water wheel the first maximum is at and a quarter period is , so the rising crossing is at . This gives . It draws the same graph as the cosine version. (p. 397)
Both answers are correct. Pick the one whose shift is easiest to read from the graph.
Suppose a graph has range and a period of .
With no shift, the sine function is .
The sine curve reaches its first maximum a quarter period in, at . A cosine curve has its maximum at its starting point, so shift it right by . This gives , which is the same graph. (p. 395)
Once you know what each parameter does, you can read an equation as a description of a situation.
Blood pressure. Take , with pressure in millimetres of mercury and in seconds.
Because is negative, the graph starts at its minimum, 80, not its maximum. (p. 395)
Hours of daylight. The model is sinusoidal because daylight repeats every year. The period is 365 days. The number 12 is the axis, so it is the average number of daylight hours.
(p. 395)
To see a full cycle on a graphing calculator, your window must cover two things:
For , the period is , so shows one cycle. The axis is and the amplitude is , so the heights run from to . A y-window such as works. Many windows are acceptable, as long as they show at least one full period and the full range. (p. 394)
A sinusoidal model has the form or . To build one, find the features in this order: axis, amplitude, period, then the shift.
A cosine model is usually easier when the graph starts at a maximum. Then is the x-coordinate of a maximum, because peaks at . A sine model needs to be where the curve crosses its axis going upward (pp. 398, 400–401).
Example. Suppose the amplitude is 2, the period is and there is a maximum at . The minimum is , so the axis is . Also . A cosine model needs no shift: . The same curve as a sine needs , the spot where it rises through the axis: (p. 398).
Fitting data. Plot the points and draw a smooth wave. For the Moon-visibility data, the maximum is 1, the minimum is 0 and the pattern repeats every 30 days. So:
The model is . On day 110 the angle is , so . About 65% of the Moon is visible (pp. 399–400).
Restricting domain and range. A real situation can limit the inputs and outputs. The Moon model uses because is a day of the year, and its range is . A swaying flagpole only makes sense for (pp. 400–401, 407).
Half a period. The time from a maximum to the next minimum is half a period. John's Ferris wheel height is 11 m at 10 s and 1 m at 55 s. That gap is 45 s, so the period is 90 s and . The axis is and the amplitude is 5. The first maximum is at 10 s, so (p. 404).
At 78 s, m. Check that this fits the graph (pp. 404–405).
Matching changes to parameters. Each change in the situation alters one parameter:
If both the faster turn and the higher axle apply, becomes 6 and becomes 7 (pp. 405, 407).
Starting from words. Sketch a rough graph first. Chantelle on a trapeze is 12 m from a beam at 1 s (a maximum) and 4 m at 3 s (a minimum). The axis is 8 and the amplitude is 4. The period is s, so . The model is (pp. 408–409).
Flagpole. The top sways 10 cm each way, 240 times per minute, and starts at rest moving right.
If the sway is cut by 20% with the same period, the amplitude is . Only changes: (p. 407).
Meanings in context. The axis is the average or resting value. The amplitude is the greatest displacement from it. The period is the time for one full cycle. For a swinging trailer tail light, these are the average distance from the curb, the largest swing to either side, and the time for one full swing back and forth (p. 408).
Misconception. Changing or does not change the amplitude or the axis. In , the amplitude is 3 and the axis is whatever and are. Only sets the amplitude and only sets the axis (p. 420).
The notes of a piano can each be modelled by , and a larger means a higher pitch. Adding the functions for several notes, such as the A major chord, gives a wave that repeats. So it is periodic, but its shape is more complicated than a single smooth sine wave, so it is not sinusoidal. A note with its octave gives a regular pattern. Dissonant combinations tend to look less regular (p. 412).
A sinusoidal curve also appears when paper wrapped around a cylinder is cut along a slanted line and unrolled. One trip around the cylinder is one cycle, so the period equals the circumference, . The axis is halfway between the two marked heights, and the amplitude is half their difference. For a cylinder with radius 7 cm and points at 12 cm and 8 cm, the axis is , the amplitude is 2 and the period is cm (p. 417).
