Chapter 2.6, Problem 48PPSE

Solution

To sketch the graph of the transformed function $y=f\left(x+2\right)+1$ from the graph of the function $y=f\left(x\right)$ .

Given:

Given the graph of $y=f\left(x\right)$:

  

Concept Used:

Horizontal Translation:

Given the graph of a function $y=f\left(x\right)$ .

The transformation $y=f\left(x-k\right)$ translates $y=f\left(x\right)$ horizontally right by $k$ units if $k>0$ .

The transformation $y=f\left(x-k\right)$ translates $y=f\left(x\right)$ horizontally left by $k$ units if $k<0$ .

Vertical Translation:

Given the graph of a function $y=f\left(x\right)$ .

The transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ vertically up by $k$ units if $k>0$ .

The transformation $y=f\left(x\right)+k$ translates $y=f\left(x\right)$ vertically down by $k$ units if $k<0$ .

Graph:

Translate the graph of $y=f\left(x\right)$ horizontally left by $2$ units by applying the transformation $y=f\left(x+2\right)$:

  

Translate the graph of $y=f\left(x+2\right)$ vertically up by $1$ unit by applying the transformation $y=f\left(x+2\right)+1$:

  

Interpretations:

The graph of $y=f\left(x+2\right)+1$ is obtained form the graph of $y=f\left(x\right)$ by translating it horizontally left by $2$ units and then by translating it vertically up by $1$

unit: