Chapter 1.6, Problem 56E

(a)

To determine

To plot:The graph of function $y=f\left(x\right)$ and describe it.

Explanation of Solution

Given information:The function is $f\left(x\right)=\sin x+\cos x$ .

Graph:

To plot the graph of function $f\left(x\right)=\sin x+\cos x$ , use graphing calculator and follow the steps given below.

Step 1: Press “ON” button to open the calculator. Press $\boxed{\text{Y}=}$ to plot the graph of a function.

Step 2: Write the right hand side of the function $f\left(x\right)=\sin x+\cos x$ after the symbol ${\text{Y}}_1$ . Enter the keystrokes $\boxed{\text{SIN}}\boxed{\text{X}}\boxed{)}\boxed{+}\boxed{\text{COS}}\boxed{\text{X}}\boxed{)}$ .

Step 3: Press $\boxed{\text{WINDOW}}$ and set the window at $\left[-2\pi ,2\pi \right]$ by $\left[-2,2\right]$ . Press the $\boxed{\text{GRAPH}}$ key to produce the graph as shown below.

  

Figure (1)

Interpretation: From the above graph it can be observed that it is a graph of sine or cosine function with shifting and amplitude will be greater than 1.

(b)

To determine

To find: The amplitude, period, horizontal shift and vertical from the graph of the function.

Answer to Problem 56E

For the function $f\left(x\right)=\sin x+\cos x$ the amplitude is $\sqrt{2}$ , period is $2\pi$ , horizontal shift is $\frac{\pi }{4}$ to the left and no vertical shift.

Explanation of Solution

Given information:The function is $f\left(x\right)=\sin x+\cos x$ .

Calculation:

From part (a), the graph of the function in the window $\left[-2\pi ,2\pi \right]$ by $\left[-2,2\right]$ .

  

The period is the distance required to complete one cycle of a function. From the above graph, it can be observed that one cycle is completed from $-2\pi$ to $2\pi$ . So, the period of the function is $2\pi$ .

The height from the center line to the peak of the graph shows the amplitude of the function. So, the amplitude of the given function is $\sqrt{2}$ .

The upwards or downwards shifting of a graph is known as the vertical shift. From the graph it can be observed that there is no vertical shift of the graph.

The left or right shifting of a graph is known as the horizontal shift. From the graph it can be observed that the graph of the given function is shifted $\frac{\pi }{4}$ to the left.

Therefore, the amplitude is $\sqrt{2}$ , period is $2\pi$ , horizontal shift is $\frac{\pi }{4}$ to the left and no vertical shift for the function $f\left(x\right)=\sin x+\cos x$ .