Chapter 3.5, Problem 61PFA

To determine

To find: Transformed function.

Answer to Problem 61PFA

Transformed function is $g\left(x\right)=-x+3$

Explanation of Solution

Given information:

  $f\left(x\right)=x$ is reflected across the X-axis and translated up 3 units.

Concept:

A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.

The graph of $y=f\left(x\right)$ where $f\left(x\right)=x$ is same as the graph of $y=x$ . Writing the equations as functions in the form of $f\left(x\right)$ is useful when applying translations and reflections to graphs.

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

If $g\left(x\right)=f\left(x+l\right)$ it is known as horizontal shift.

If $l<0$ moves it right.

If $l>0$ moves it left.

If $g\left(x\right)=-f\left(x\right)$ it is known as Vertical reflection it means reflects about X-axis

If $g\left(x\right)=f\left(-x\right)$ it is known as Horizontal reflection it means reflects about Y-axis

If $g\left(x\right)=af\left(x\right)$

If $a>1$ it is known as Vertical stretch.

If $0<a<1$ it is known as Vertical compression.

If $g\left(x\right)=f\left(bx\right)$

If $b>1$ it is known as Horizontal compression.

If $0<b<1$ it is known as Horizontal Stretch.

Calculation:

Given first $f\left(x\right)=x$ is reflected across the X-axis

If $g\left(x\right)=-f\left(x\right)$ it is known as Vertical reflection it means reflects about X-axis

Therefore

We get

  $\begin{array}{l}\Rightarrow g\left(x\right)=-f\left(x\right)\\ \Rightarrow g\left(x\right)=-x\end{array}$

Then it is translated up 3 units

If $g\left(x\right)=f\left(x\right)+k$ it is known as vertical shift.

If $k>0$ it moves upwards.

If $k<0$ it moves downwards.

Here as given it translates up $k>0$

We get

  $k=3$

Therefore

  $\begin{array}{l}\Rightarrow g\left(x\right)=-f\left(x\right)+k\\ \Rightarrow g\left(x\right)=-x+3\end{array}$

Therefore, the transformed function is $g\left(x\right)=-x+3$ .