Question

How do you find the amplitude, period, vertical and phase shift and graph for $y=\cos \left( x-{{45}^{\circ }} \right)$?

Answer 1

Convert the given angle in degrees into radian and write the function as $y=\cos \left( x-\dfrac{\pi }{4} \right)$. Assume $y=\cos x$ as the reference function. Now, compare the given function $y=\cos \left( x-\dfrac{\pi }{4} \right)$ with the general form: $y=a\cos \left[ b\left( x+c \right) \right]+d$. Find the corresponding values of a, b, c and d and using these values find the vertical compression or stretch (a), also known as the amplitude, horizontal compression or stretch (b), horizontal shift (c), also known as the phase shift, and vertical shift (d) of the function. Now, to find the period of the function use the formula: $T'=\dfrac{T}{\left| b \right|}$, where T is the period of $\cos x$. Finally, draw the graph of the given function.

Complete step by step solution:
Here, we have been provided with the function $y=\cos \left( x-{{45}^{\circ }} \right)$ and we are asked to find its amplitude, time period and vertical and phase shift and we have to draw its graph. Generally we assume $y=\cos x$ as our reference function. We know that ${{45}^{\circ }}=\dfrac{\pi }{4}$ radian, so we have the function as $y=\cos \left( x-\dfrac{\pi }{4} \right)$.
Now, the general form of the cosine function is given as $y=a\cos \left[ b\left( x+c \right) \right]+d$. Here, we have the vertical compression or stretch (a), also known as the amplitude, horizontal compression or stretch (b), horizontal shift (c), also known as the phase shift, and vertical shift (d) of the function. The descriptions for this transform are:
1. If $\left| a \right|>1$ then vertical stretch takes place and if 0 < a < 1 then vertical compression takes place.
2. If ‘a’ is negative then the function is reflected about the x – axis.
3. If $\left| b \right|>1$ then horizontal stretch takes place and if 0 < b < 1 then horizontal compression takes place.
4. If ‘b’ is negative then the function is reflected about y – axis.
5. If ‘c’ is negative then the function is shifted $\left| c \right|$ units to the right and if ‘c’ is positive then the function is shifted $\left| c \right|$ units to the left.
6. If ‘d’ is negative then the function is shifted $\left| d \right|$ units down and if ‘d’ is positive then the function is shifted $\left| d \right|$ units up.
Now, on comparing the given function $y=\cos \left( x-\dfrac{\pi }{4} \right)$ with the general form $y=a\cos \left[ b\left( x+c \right) \right]+d$, we have,
$\Rightarrow a=1,b=1,c=-\dfrac{\pi }{4}$ and $d=0$.
(i) Amplitude: - Amplitude of a function is defined as the minimum and maximum value that a function can approach. Clearly, the value of ‘a’ is 1 unit. Therefore the amplitude of the given function is 1 unit. Therefore, there will not be any vertical stretch or compression in the function provided.
(ii) Time period: - Time period of a function is defined as the interval of x after which the value of the function starts repeating itself. It is also simply called the period and the function is known as periodic function. The period of the function $y=a\cos \left[ b\left( x+c \right) \right]+d$ is given as $T'=\dfrac{T}{\left| b \right|}$, where T is the period of $\cos x$.
Now, we know that the period of $\cos x$ is $T=2\pi$, so we have,

& \Rightarrow T'=\dfrac{2\pi }{\left| 1 \right|} \\\ & \Rightarrow T'=2\pi \\\ \end{aligned}$$ Therefore, the period of the provided function is $2\pi $. (iii) Phase shift: - Phase shift of a function is defined as how far a function is shifted horizontally from the usual position. Clearly, the value of c is $-\dfrac{\pi }{4}$, so the phase shift is $-\dfrac{\pi }{4}$. Therefore, the function will be shifted $$\dfrac{\pi }{4}$$ units to the right. (iv) Vertical shift: - Vertical shift of a function is defined as how far a function is shifted vertically from the usual position. Clearly, the value of d is 0, so the vertical shift is 0 units. Therefore, the function will not be shifted vertically. Now, let us draw the graph of the function.  **Note:** You must remember the descriptions of the transform of the function otherwise it will be difficult to draw the graph. Note that the phase shift has no effect on time period. Time period changes only when the coefficient of x is changed in the function. The graphs of these functions are helpful in chapters such as integration where we have to find the area under the curve that is a part of a graph.