What are the four transformations of a trigonometric graph?

The four are a change of amplitude (a vertical stretch), a vertical translation , a multiple angle (a horizontal stretch) and a phase angle (a horizontal translation). The first two are vertical changes and the last two horizontal ones.

Which transformations leave the amplitude alone, and which leave the period alone?

Only a change of amplitude affects the amplitude, and only a multiple angle affects the period. A vertical translation and a phase angle both leave the amplitude and the period exactly as they were.

True or False? When a vertical and a horizontal transformation are combined, the order they are applied in matters.

False. Vertical and horizontal transformations are independent , so combining one of each gives the same graph whichever order you use. Two transformations of the same sort do affect each other, and those must be applied in the right order.

National 5MathsSQAFlashcardsTrigonometric SkillsGraphs of Trigonometric Functions

Graphs of Trigonometric Functions (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontGraphs of Basic Trigonometric Functions
Complete the periods of the three basic trigonometric graphs.
$y = \sin x repeats every______$
$y = \cos x repeats every______$
$y = \tan x repeats every______$

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Cards in this collection (24)

Complete the periods of the three basic trigonometric graphs.
$y = \sin x repeats every______$
$y = \cos x repeats every______$
$y = \tan x repeats every______$
The completed periods are:
$y = \sin x and y = \cos x repeat every 360 °$
$y = \tan x repeats every 180 °$
The period of $\tan x$ is half that of the other two.
For $y = \sin x$ what is the amplitude and where does the curve start?
The amplitude is $1$ , so the wave rises $1$ above and falls $1$ below the $x$ axis.
It passes through the origin, and then every $90 °$ it takes the heights $1$ , $0$ , $- 1$ , $0$ in turn.
For $y = \cos x$ what is the amplitude and where does the curve start?
The amplitude is $1$ , exactly as for the sine wave.
The curve starts at its maximum, with a $y$ intercept of $1$ , and then every $90 °$ takes the heights $0$ , $- 1$ , $0$ , $1$ in turn.
How does the graph of $y = \cos x$ relate to the graph of $y = \sin x$ itself?
The cosine graph is the sine graph translated $90 °$ to the left.
That is why the two share a shape, an amplitude and a period, yet start at different heights.
True or False?
$y = \tan x$ oscillates between a maximum of $1$ and a minimum of $- 1$ .
False.
The tangent graph has no maximum or minimum value at all, and can take any value, positive, negative or zero.
It is not a wave either: it consists of separate branches running off towards $+ \infty$ and $- \infty$ .
Define an asymptote of a graph.
An asymptote is a line that the curve gets closer and closer to but never touches.
On $y = \tan x$ they occur every $180 °$ , at $x = 90 °$ , $x = 270 °$ and so on, and are usually drawn dotted.
Which way does a branch of $y = \tan x$ run near an asymptote?
On the left of an asymptote the branch climbs towards $+ \infty$ , and on the right of one it falls from $- \infty$ .
Each branch therefore rises from bottom left to top right, crossing the $x$ axis once between two asymptotes.
What values can $x$ take in a trigonometric graph?
Any value at all, since the angle is not limited to acute angles.
It may be obtuse, reflex, negative, or greater than $360 °$ , which is why the graphs repeat in both directions.
What are the four transformations of a trigonometric graph?
The four are a change of amplitude (a vertical stretch), a vertical translation, a multiple angle (a horizontal stretch) and a phase angle (a horizontal translation).
The first two are vertical changes and the last two horizontal ones.
What does the $a$ in $y = a \sin x$ change?
The amplitude becomes $a$ , so the wave oscillates between $a$ and $- a$ instead of $1$ and $- 1$ .
Its turning points move to $(90 °, a)$ and $(270 °, - a)$ , while the roots stay at $0 °$ , $180 °$ and $360 °$ .
True or False?
A vertical translation can leave a sine graph with no roots at all.
True.
The wave oscillates between $c + 1$ and $c - 1$ , so if $c$ is bigger than $1$ the whole curve sits above the $x$ axis.
The graph of $y = \sin x + 2$ runs between $1$ and $3$ and never crosses the axis at all.
What does the $c$ in $y = \sin x + c$ change?
The middle line moves from the $x$ axis up to $y = c$ , so the wave now runs between $c + 1$ and $c - 1$ .
The amplitude is still $1$ , and a positive $c$ shifts the graph up while a negative $c$ shifts it down.
How does $y = \sin x + c$ differ from $y = \sin (x + c)$ in effect?
In $y = \sin x + c$ the $c$ is outside the function, so it moves the graph vertically.
In $y = \sin (x + c)$ it is inside, so it is a phase angle and moves the graph horizontally instead.
Complete the period of a graph with a multiple angle.
$y = \sin b x has period______$
The completed period is:
$y = \sin b x has period (\frac{360}{b}) °$
A $b$ greater than $1$ squeezes more cycles into $360 °$ , while a $b$ between $0$ and $1$ stretches the graph out.
For $y = \sin (x + d)$ which way does a positive $d$ move the graph?
A positive $d$ moves the graph to the left, which is the opposite of what the plus sign suggests.
The starting point moves from the origin to $(- d, 0)$ , and a negative $d$ moves the graph to the right.
Which transformations leave the amplitude alone, and which leave the period alone?
Only a change of amplitude affects the amplitude, and only a multiple angle affects the period.
A vertical translation and a phase angle both leave the amplitude and the period exactly as they were.
True or False?
When a vertical and a horizontal transformation are combined, the order they are applied in matters.
False.
Vertical and horizontal transformations are independent, so combining one of each gives the same graph whichever order you use.
Two transformations of the same sort do affect each other, and those must be applied in the right order.
For $y = a \sin b x$ where does each feature come from?
The $a$ gives the amplitude and the $y$ coordinates of the turning points, exactly as in $y = a \sin x$ .
The $b$ gives the period and the $x$ coordinates of the roots and turning points, exactly as in $y = \sin b x$ .
For $y = 2 \sin x - 4$ in which order are the two changes applied?
The change of amplitude comes first, so the graph is stretched by a factor of $2$ and then translated $4$ units down.
To find a point, multiply the $y$ coordinate by $2$ and then subtract $4$ , leaving the $x$ coordinates alone.
For $y = \cos (2 x + 30)$ in which order are the two changes applied?
The phase angle comes first, so the graph is translated $30 °$ left and then stretched by a factor of $\frac{1}{2}$ .
To find a point, subtract $30$ from the $x$ coordinate and then divide by $2$ , leaving the $y$ coordinates alone.
For $y = \cos (2 x + 30)$ complete the coordinates of the first minimum turning point.
$(\frac{180 -______}{2}, - 1) = (______, - 1)$
The completed coordinates are:
$(\frac{180 - 30}{2}, - 1) = (75, - 1)$
The minimum of $\cos x$ sits at $180 °$ , and that $x$ coordinate is adjusted by subtracting the $30$ and then halving.
How do you find the equation of a transformed trigonometric graph?
Compare features you can read off the graph, such as the amplitude, the period or a turning point, with what they should be in terms of the letters.
A maximum at $(0, 7)$ with two cycles in $360 °$ gives $a = 7$ and $\frac{360}{b} = 180$ , so $y = 7 \cos 2 x$ .
How do you find a point's coordinates from a transformed equation?
Start from where that point sits on the basic graph, then apply each transformation to its coordinates in turn.
For $y = 3 \sin (x - 60)$ the first maximum of $\sin x$ at $(90, 1)$ moves to $(150, 3)$ .
A graph of $y = \sin (x + a) + b$ has its minimum at $(210, 0)$ so find $a$ and $b$ from it.
The minimum of $y = \sin x$ is at $(270, - 1)$ , so this one has moved $60 °$ left and $1$ unit up.
A shift to the left means $a$ is positive, giving $a = 60$ and $b = 1$ .

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