Question

What is the amplitude of $y = \cos \left( { - 3x} \right)$ and how does the graph relate to $y = \cos x$?

Answer 1

We know that cosine is the periodic function. We will first use the general form equation of the cosine function to find the amplitude and period of both the functions. After that, we will see how the graph of $y = \cos \left( { - 3x} \right)$relates to $y = \cos x$ using these values.

Formula used:
$y = A\cos \left( {Bx + C} \right) + D$,where, $A$ is the vertical stretch factor whose absolute value is the amplitude, $B$is used to find the period $P = \dfrac{{2\pi }}{B}$, $C$ is the place shift and $D$ is the vertical shift.

Complete step by step answer:
We are asked to find the amplitude of the function $y = \cos \left( { - 3x} \right)$.For this, we will use the general equation of the cosine function and compare it with the given one to find its amplitude and period.The general form of cosine equation is: $y = A\cos \left( {Bx + C} \right) + D$.If we compare it with the given function, we get $A = 1$ and $B = - 3$. Other two values are not given in the equation.Thus the amplitude of $y = \cos \left( { - 3x} \right)$ is 1.And the period of $y = \cos \left( { - 3x} \right)$ is $P = \dfrac{{2\pi }}{B} = - \dfrac{{2\pi }}{3}$ we can consider it as $\dfrac{{2\pi }}{3}$ because cos is even function.

Now, we will do the same for the standard function $y = \cos x$.
For this, we get $A = 1$and $B = 1$
Therefore, the amplitude of $y = \cos x$ is 1.
And the period of $y = \cos x$ is $P = \dfrac{{2\pi }}{1} = 2\pi$.
From this, we can say that as the amplitude of $y = \cos \left( { - 3x} \right)$ and $y = \cos x$ is the same, their range will be the same from $- 1$ to $1$.
However, the period of $y = \cos \left( { - 3x} \right)$ is less than that of $y = \cos x$.

Therefore, the phase difference between the graphs of these functions will be $2\pi - \dfrac{{2\pi }}{3} = \dfrac{{4\pi }}{3}$.This means that both the graphs intersects each other after the period of $\dfrac{{4\pi }}{3}$.

Note: Here, we have determined the amplitude and period of the given cosine functions. The Amplitude is defined as the height from the center line to the peak or to the trough. Also, we can measure the height from the highest to lowest points and divide that value by 2 to get the value of the amplitude. A Periodic Function is a function that repeats its values in regular intervals or periods. For example, both the functions given here, have a certain period which makes them periodic functions.