Define the amplitude of an oscillating graph.

The amplitude is half the vertical distance between the maximum and the minimum points of the graph. It is also the distance from the graph's horizontal line of symmetry up to a maximum point.

Define the period of an oscillating graph.

The period is the distance along the x -axis taken to complete one full oscillation . You can measure it from one maximum point to the next, or from one minimum point to the next.

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Trigonometric Graphs & Equations (Cambridge (CIE) IGCSE International Maths: Extended): Flashcards

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What is the graph of $y = \sin x$ for $- 360 ° \leq x \leq 360 °$ ?

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What is the graph of $y = \sin x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \sin x$ for $- 360 ° \leq x \leq 360 °$ is:
What is the graph of $y = \cos x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \cos x$ for $- 360 ° \leq x \leq 360 °$ is:
What is the graph of $y = \tan x$ for $- 360 ° \leq x \leq 360 °$ ?
The graph of $y = \tan x$ for $- 360 ° \leq x \leq 360 °$ is:
True or False?
The point $(0, 0)$ lies on the graph $y = \sin x$ .
True.
The point $(0, 0)$ lies on the graph $y = \sin x$ .
What is the y-intercept of the graph $y = \cos x$ ?
The y-intercept of the graph $y = \cos x$ is $(0, 1)$ .
What is the minimum y value of the graph $y = \sin x$ ?
The minimum y value of the graph $y = \sin x$ is -1.
True or False?
The graph $y = \cos x$ repeats itself every 180°.
False.
The graph $y = \cos x$ does not repeat itself every 180°.
It repeats itself every 360°.
True or False?
The graph $y = \tan x$ repeats itself every 180°.
True.
The graph $y = \tan x$ repeats itself every 180°.
The graph $y = \sin x$ repeats itself every how many degrees?
The graph $y = \sin x$ repeats itself every 360°.
True or False?
The maximum y value on the graph $y = \tan x$ is 1.
False.
The maximum y value on the graph $y = \tan x$ is not 1.
The graph $y = \tan x$ does not have a maximum value.
Define the amplitude of an oscillating graph.
The amplitude is half the vertical distance between the maximum and the minimum points of the graph.
It is also the distance from the graph's horizontal line of symmetry up to a maximum point.
Define the period of an oscillating graph.
The period is the distance along the x-axis taken to complete one full oscillation.
You can measure it from one maximum point to the next, or from one minimum point to the next.
In the equation $y = a \sin (b x)$ what does a control, and what does b control?
a is the amplitude, so it sets how far the graph reaches above and below its horizontal line of symmetry.
b does not affect the amplitude at all: it changes the period instead.
A sine or cosine graph can be written as $y = a \sin (b x)$ or as $y = a \cos (b x)$ for some values of a and b. Complete the formula for its period:
$period = \frac{______}{b}$
The completed formula is:
$period = \frac{360 °}{b}$
This works for both sine and cosine graphs, but not for $y = \tan x$ which repeats twice as often.
True or False?
On the graph of $y = a \sin (b x)$ the distance between two neighbouring x-intercepts is one full period.
False.
That distance is only half a period.
A full period runs from one x-intercept to the next but one, because the graph crosses the axis twice in every cycle, once going up and once coming down.
On the graph of $y = a \sin (b x)$ what happens as b gets bigger?
The period gets shorter, so the graph is squashed horizontally and fits more cycles into the same interval.
A larger b makes the graph repeat more often; it does not make the graph taller.
The graph $y = 2 + \frac{1}{5} \cos x$ oscillates about the line $y = 2$ rather than the x-axis. What is its amplitude?
Its amplitude is $\frac{1}{5}$ of a unit.
Amplitude is measured from the horizontal line of symmetry to a maximum or minimum point, so adding 2 slides the whole graph upwards without changing how far it swings.
A graph of the form $y = a \sin (b x)$ has a maximum of 3, a minimum of $- 3$ and repeats every $90 °$ along the x-axis. Find a and b.
a is the amplitude, which is 3 here, since the graph reaches 3 above and 3 below the x-axis.
The period is $90 °$ which is a quarter of a full $360 °$ turn, so the graph repeats four times as often and b is 4:
$y = 3 \sin (4 x)$
How would you use a graph of $y = \sin x$ to find the solutions of $\sin x = 0.5$ for $0 ° \leq x \leq 360 °$ ?
To find the solutions of $\sin x = 0.5$ for $0 ° \leq x \leq 360 °$ using the graph $y = \sin x$ :
- calculate one solution using inverse trig $x = \sin^{- 1} (0.5)$
- draw the horizontal line $y = 0.5$
- use the symmetry of the graph to find the other solution
True or False?
After finding the first solution for an equation involving the cosine function, you can find another solution by subtracting the first solution from 180º.
False.
After finding the first solution for an equation involving the cosine function, you can find another solution by subtracting the first solution from 360º.
What angle should you add to or subtract from a first solution to find another solution for an equation involving the tangent function?
If you know a first solution for an equation involving the tangent function, you can add to or subtract 180º from it to find another solution .

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