MCR3U Graphs of y = sin x and y = cos x — Transformations: Notes and Practice

$y = sin x$ and $y = cos x$ are the two 'parent' periodic curves everything else in this unit is built from. Both wave smoothly between $- 1$ and $1$ forever, repeating every $360°$ — they're identical in shape, just horizontally shifted from each other by $90°$ . Sound waves, alternating current in your home's wiring, and the swing of a pendulum all trace out stretched and shifted copies of these two curves. If you know the parent graphs cold, and you know what each of the four parameters $a$ , $k$ , $d$ and $c$ does to them, you can sketch any sinusoid in under a minute and read an equation straight off a graph.

Fig 1y = sin x starts at the axis and rises; y = cos x starts at its maximum.

Feature	$y = sin x$	$y = cos x$
Value at $x = 0$	$0$	$1$
Maximum	$1$ (at $x = 90°$ )	$1$ (at $x = 0°$ )
Minimum	$- 1$ (at $x = 270°$ )	$- 1$ (at $x = 180°$ )
Zeros from $0°$ to $360°$	$0°, 180°, 360°$	$90°, 270°$
Period	$360°$	$360°$
Amplitude	$1$	$1$
Axis	$y = 0$	$y = 0$

Why do the graphs look like this? Picture a point travelling around the unit circle. Its $y$ -coordinate is $sin x$ and its $x$ -coordinate is $cos x$ . Starting at $(1, 0)$ , the height is $0$ , climbs to $1$ at $90°$ (the top of the circle), falls back to $0$ at $180°$ , reaches $- 1$ at $270°$ and returns to $0$ at $360°$ , where the point is back where it started and the pattern repeats. That is the sine graph. The $x$ -coordinate starts at $1$ , so the cosine graph starts at its maximum. These five KEY POINTS, spaced a quarter-period ( $90°$ ) apart, are all you need to sketch either curve: sine goes axis, max, axis, min, axis; cosine goes max, axis, min, axis, max. Because the cosine curve reaches each feature $90°$ earlier, $cos x = sin (x + 90°)$ : cosine is simply sine shifted $90°$ to the left.

What each parameter does

y = a sin (k (x - d)) + c

$a$ — vertical stretch/compression; $∣ a ∣$ becomes the new AMPLITUDE, because every $y$ -value of the parent (which runs from $- 1$ to $1$ ) is multiplied by $a$ . If $a < 0$ , the graph reflects vertically (flips upside down).
$k$ — horizontal stretch/compression; the new PERIOD becomes $\frac{360°}{∣ k ∣}$ . A larger $∣ k ∣$ squishes the graph (shorter period, faster oscillation). Why: one cycle ends when the angle INSIDE the sine reaches $360°$ , and $k x = 360°$ happens at $x = \frac{360°}{k}$ .
$d$ — horizontal (phase) shift; the graph slides $d$ units right if $d > 0$ , left if $d < 0$ . Why: the inside, $k (x - d)$ , equals $0$ when $x = d$ , so the start of the cycle moves from $0$ to $d$ .
$c$ — vertical shift; becomes the new AXIS (midline) of the graph, $y = c$ , because $c$ is added to every $y$ -value.

amplitude = ∣ a ∣ period = \frac{360°}{∣ k ∣} axis: y = c max = c + ∣ a ∣, min = c - ∣ a ∣

Read d only after factoring k

Watch the SIGN inside the brackets:

y = sin (k (x - d))

shifts RIGHT when

d

is positive. A formula like

y = sin (x + 30°)

must first be rewritten as

y = sin (x - (- 30°))

to correctly read off

d = - 30°

(a shift LEFT). And if

k

has not been factored out, do that first:

y = sin (2 x - 60°) = sin (2 (x - 30°)),

so the phase shift is

30°

right, NOT

60°

Graphing in order: the transformation sequence

Start with the five key points of the parent: for sine, $(0°, 0)$ , $(90°, 1)$ , $(180°, 0)$ , $(270°, - 1)$ , $(360°, 0)$ .
Apply the stretches and reflections first: divide each $x$ -coordinate by $k$ , and multiply each $y$ -coordinate by $a$ .
Then apply the translations: add $d$ to each $x$ -coordinate and add $c$ to each $y$ -coordinate.
In one line, the mapping rule is $(x, y) \to (\frac{x}{k} + d, a y + c)$ .
Plot the new points, join them with a smooth wave, and repeat the cycle every period in both directions.

Parent point $(x, y)$	$(\frac{x}{2} + 30°, 3 y + 5)$	Feature
$(0°, 0)$	$(30°, 5)$	On the axis, rising
$(90°, 1)$	$(75°, 8)$	Maximum
$(180°, 0)$	$(120°, 5)$	On the axis, falling
$(270°, - 1)$	$(165°, 2)$	Minimum
$(360°, 0)$	$(210°, 5)$	End of one cycle

Key points of y = 3sin(2(x − 30°)) + 5: one cycle from 30° to 210°, a period of 180°.

Loading interactive…

Drag a, k, d and c and watch the sine curve stretch, compress, shift, and rise or fall in real time.

VideoGraphs transformed sine and cosine functions by sending the parent's key points through the mapping rule: divide x by k and add d, multiply y by a and add c, in the same order as this section.Watch on YouTube

Writing an equation from a graph

Reverse the process. Read the maximum and minimum: $c = \frac{m a x + m i n}{2}$ and $∣ a ∣ = \frac{m a x - m i n}{2}$ . Measure the period (peak to peak, or twice the distance from a peak to the next trough) and compute $k = \frac{360°}{period}$ . Then choose a starting point for $d$ . For a cosine equation, $d$ is the $x$ -value of any MAXIMUM, since cosine begins its cycle at a maximum. For a sine equation, $d$ is the $x$ -value where the graph crosses the axis going UP. Using a minimum with cosine also works if you make $a$ negative. Because the graph repeats, there are infinitely many correct equations (you can add any whole number of periods to $d$ ), so on a test any correct one earns full marks; check your answer by substituting one point from the graph.

Why there is never just one right equation

Take the curve with a maximum at

(45°, 7)

, a minimum at

(135°, - 1)

and period

180°

. Starting the cosine cycle at the maximum gives

y = 4 cos (2 (x - 45°)) + 3

; starting it one period later gives

y = 4 cos (2 (x - 225°)) + 3

, the same curve. Starting at the minimum instead and flipping gives

y = - 4 cos (2 (x - 135°)) + 3

, again the same curve, because the minimum is half a period from the maximum. And the sine version

y = 4 sin (2 x) + 3

uses the rising axis crossing at

x = 0°

. To decide whether two equations match, compare their amplitude, period and axis, then test one point such as

x = 45°

in each: all four give

y = 7

there.

Graphs of y = sin x and y = cos x — Transformations

Key things to remember

What each parameter does

Graphing in order: the transformation sequence

Writing an equation from a graph

Worked examples

Where marks get lost

Flashcards 10

Practice