Chapter 3, Problem 2SGA

To determine

To find: whether graph can trace without lifting the pencil and graph will be continuous or discontinuous.

Answer to Problem 2SGA

The graph can be trace without lifting the pencil.

Explanation of Solution

Given:

Graph of the function and pencil.

Concept used:

It knows that the graph of the function $y=f\left(x\right)+a$ can be obtained from the parent function

$y=f\left(x\right)$ is translate it $a$ units up.

It knows that the graph of the function $y=f\left(x-a\right)$ can be obtained from the parent function

$y=f\left(x\right)$ is translate it $a$ units right.

And the graph of the function $y=-f\left(x\right)$ can be obtained from the parent function $y=f\left(x\right)$ by reflecting over $x-$ axis.

It knows that the graph of the function $y=cf\left(x\right)$ can be obtained from the parent function $y=f\left(x\right)$ by compressing the graph vertically with a factor $c$ .

Calculation:

Let’s consider the parent function $f\left(x\right)$ :

It knows that the graph of the function $y=f\left(x\right)+a$ can be obtained from the parent function

$y=f\left(x\right)$ is translate it $a$ units up.

It knows that the graph of the function $y=f\left(x-a\right)$ can be obtained from the parent function

$y=f\left(x\right)$ is translate it $a$ units right.

And the graph of the function $y=-f\left(x\right)$ can be obtained from the parent function $y=f\left(x\right)$ by reflecting over $x-$ axis.

It knows that the graph of the function $y=cf\left(x\right)$ can be obtained from the parent function $y=f\left(x\right)$ by compressing the graph vertically with a factor $c$ .

Therefore ,the graph of the function $y=-2f\left(x-4\right)+1$ can be obtained from the parent function $f\left(x\right)$ by translate it $4$ unit right, reflecting over $x$ - axis, expanding vertically $2$ by a factor of $2$ , and then translate $1$ unit up.

For example, if the parent function is $f\left(x\right)=x^2$ the required function is:

  $f\left(x\right)=-2{\left(x-2\right)}^2+1$ .

Hence, graph can be trace without lifting the pencil.