Chapter 8, Problem 8CR

(a)

To determine

To graph: The function $y=x$

Explanation of Solution

Given information:

The function $y=x$

Graph:

To graph the function $y=x$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=x\\ -2& -2\\ 0& 0\\ 2& 2\end{array}$

Plot these points on graph and draw a line passing through it.

The graph of the function is

  

Interpretation:

The graph represents function $y=x$

(b)

To determine

To graph: The function $y=x^2$

Explanation of Solution

Given information:

The function $y=x^2$

Graph:

To graph the function $y=x^2$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=x^2\\ -3& 9\\ -2& 4\\ 0& 0\\ 2& 4\\ 3& 9\end{array}$

Plot these points on graph and draw a curve passing through it.

The graph of the function is

  

Interpretation:

The graph represents function $y=x^2$

(c)

To determine

To graph: The function $y=\sqrt{x}$

Explanation of Solution

Given information:

The function $y=\sqrt{x}$

Graph:

To graph the function $y=\sqrt{x}$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=\sqrt{x}\\ 4& 2\\ 1& 1\\ 9& 3\end{array}$

Plot these points on graph and draw a curve passing through it.

The graph of the function is

  

Interpretation:

The graph represents function $y=\sqrt{x}$

(d)

To determine

To graph: The function $y=x^3$

Explanation of Solution

Given information:

The function $y=x^3$

Graph:

To graph the function $y=x^3$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=x^3\\ -1& -1\\ -0.75& -0.421875\\ 0& 0\\ 0.75& 0.75\\ 1& 1\end{array}$

Plot these points on graph and draw a curve passing through it.

The graph of the function is

  

Interpretation:

The graph represents function $y=x^3$

(e)

To determine

To graph: The function $y=e^x$

Explanation of Solution

Given information:

The function $y=e^x$

Graph:

To graph the function $y=e^x$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=e^x\\ -2& 0.135\\ -1& 0.368\\ 0& 1\end{array}$

Plot these points on graph and draw a curve passing through it.

The graph of the function is

  

Interpretation:

The graph represents function $y=e^x$

(f)

To determine

To graph: The function $y=\ln x$

Explanation of Solution

Given information:

The function $y=\ln x$

Graph:

To graph the function $y=\ln x$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=\ln x\\ 1& \ln \left(1\right)=0\\ 2& \ln \left(2\right)=0.693\\ 5& \ln \left(5\right)=1.609\end{array}$

Plot these points on graph and draw a curve passing through it.

The graph of the function is

  

Interpretation:

The graph represents the function $y=\ln x$

(g)

To determine

To graph: The function $y=\sin x$

Explanation of Solution

Given information:

The function $y=\sin x$

Graph:

To graph the function $y=\sin x$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=\sin x\\ -\pi & 0\\ -\frac{\pi }{2}& -1\\ 0& 0\\ \frac{\pi }{2}& 1\\ \pi & 0\end{array}$

Plot these points on graph and draw a curve passing through it.

Sine is periodic function with period $2\pi$

The graph of the function is

  

Interpretation:

The graph represents the function $y=\sin x$

(h)

To determine

To graph: The function $y=\cos x$

Explanation of Solution

Given information:

The function $y=\cos x$

Graph:

To graph the function $y=\cos x$, find at least three points satisfying the function equation.

  $\begin{array}{cc}x& y=\cos x\\ -\pi & -1\\ -\frac{\pi }{2}& 0\\ 0& 1\\ \frac{\pi }{2}& 0\\ \pi & -1\end{array}$

Plot these points on graph and draw a curve passing through it.

Cosine function is periodic function with period $2\pi$

Therefore, the graph of the function in interval $\left[-\pi ,\pi \right]$ is

  

Interpretation:

The graph represents the function $y=\cos x$

(i)

To determine

To graph: The function $y=\tan x$

Explanation of Solution

Given information:

The function, $y=\tan x$

Graph:

To graph the function $y=\tan x$

Domain of $y=\tan x$ is $\left[n\pi ,\frac{\pi }{2}+n\pi \right)\cup \left(\frac{\pi }{2}+n\pi ,\pi +n\pi \right]$

That is, exclude the points from the domain which are multiple of $\frac{\pi }{2}$

Tangent function is periodic function with period $\pi$

Also, vertical asymptote of $y=\tan x$ is $\frac{\pi }{2}+n\pi$

Therefore, the graph of the function is

  

Interpretation:

The graph represents the function $y=\tan x$