Chapter 3.6, Problem 2E

To determine

To write any four types of transformations and give an example of each and describe how it affects the graph of a function.

Answer to Problem 2E

The transformations are horizontal shift, vertical shift and reflections along $x$ and $y-$ axis.

Explanation of Solution

Types of transformation are as follows:

1. Horizontal shift $c$ units to the right: $h_1(x)=f(x-c).$

Example: function $h_1(x)={\left(x-2\right)}^2$ shifts $2$ unit’s right to the base function $f(x)=x^2$

2. Horizontal shift $c$ units to the left: $h_2(x)=f(x+c).$

Example: function $h_2(x)={\left(x+2\right)}^2$ shifts $2$ unit’s left to the base function $f(x)=x^2$ .

3. Vertical shift $c$ units up: $h_3(x)=f(x)+c.$

Example: function $h(x)=x^2+2$ shifts $2$ unit’s up to the base function $f(x)=x^2$ .

4. Vertical shift $c$ units down: $h_4(x)=f(x)-c$

Example: function $h_4(x)=x^2-2$ shifts $2$ unit’s down to the base function $f(x)=x^2$ .

5. Reflection in the $x-$ axis: $h_5(x)=-f(x)$

Example: function $h_5(x)=-x^2$ reflects along the $x-$ axis to the base function $f(x)=x^2$ .

6. Reflection in the $y-$ axis: $h_6(x)=f(-x)$

Example: function $h_6(x)={\left(-x\right)}^2$ reflects along the $y-$ axis to the base function $f(x)=x^2$ .

Following graph depicts the effect of translation

  

Hence, the transformations are horizontal shift, vertical shift and reflections along $x$ and $y-$ axis.