Chapter 1.4, Problem 3E

To determine

To fill in the blanks according to the given question

Answer to Problem 3E

A reflection in the x-axis of $y=f(x)$ is represented by $h(x)=-f(x)$ ,

While a reflection in the y-axis of $y=f(x)$ is represented by $h(x)=f(|x|)$

Explanation of Solution

The best way to understand this problem is by taking an example and solving it accordingly.

Let, $f(x)=x^2-5x+6$

For reflection of x -axis:

  $\begin{array}{l}h(x)=-f(x)\\ \Rightarrow h(x)=-(x^2-5x+6)\end{array}$

Calculation for graph:

Consider $f(x)=x^2-5x+6$

Values of x	Values of f (x )
0	6
1	2
-1	12
2	0
3	0

By taking different values of x , the graph can be plotted.

Calculation for graph:

Consider $h(x)=-(x^2-5x+6)$

Values of x	Values of h (x)
0	-6
1	-2
-1	-12
2	0
3	0

By taking different values of x , the graph can be plotted.

Graph for both $f(x)$ and $h(x)$ :

  

Interpretation:

From the above graph, it is clear that $h(x)$ is the reflection of $f(x)$ about x -axis.

For reflection of y -axis:

  $\begin{array}{l}h(x)=f(|x|)\\ \Rightarrow h(x)=|x{|}^2-5|x|+6\end{array}$

Calculation for graph:

Consider $f(x)=x^2-5x+6$

Values of x	Values of f (x )
0	6
1	2
-1	12
2	0
3	0

By taking different values of x , the graph can be plotted.

Calculation for graph:

Consider $h(x)=|x{|}^2-5|x|+6$

Values of x	Values of h (x)
0	6
1	2
-1	2
2	0
3	0

By taking different values of x , the graph can be plotted.

Graph for both $f(x)$ and $h(x)$ :

  

Interpretation:

From the above graph, it is clear that $h(x)$ is the reflection of $f(x)$ about y -axis.

Conclusion:

Hence,

A reflection in the x-axis of $y=f(x)$ is represented by $h(x)=-f(x)$ ,

While a reflection in the y-axis of $y=f(x)$ is represented by $h(x)=f(|x|)$