Q. 90 - Precalculus Enhanced with Graphing Utilities

Question

The graph of a function f is illustrated in the figure.

1. Draw the graph of  $y=\left|f\left(x\right)\right|$
2. Draw the graph of  $y=f\left(\left|x\right|\right)$
   

Step-by-Step Solution

Answer

(a).  

(b).

Part (a). Step 1. Given data

The graph of the function -

Part (a). Step 2. To Find

Draw the graph of  $y=\left|f\left(x\right)\right|$

Part (a). Step 3. Explanation

When $f\left(x\right)\ge 0$, we can write that -  $y=\left|f\left(x\right)\right|=f\left(x\right)$

This means that the point which are above the X-Axis will remain unchanged.

When  $f\left(x\right)<0$, We can write that  $y=\left|f\left(x\right)\right|=-f\left(x\right)$

This means that the points below the X-Axis will be reflect in the x-axis.

Part (a). Step 4. Graph

In the graph -

The given graph of $y=f\left(x\right)$ is represented by the blue solid lines.

The new graph of   $y=\left|f\left(x\right)\right|$ is represented by the orange dotted line.

• Point which are above the X-Axis will remain unchanged.
• Point below the X-Axis will be reflected in the X-Axis.

Part (b). Step 1. Given data

The graph of the function -

Part (b). Step 2. To Find

Draw the graph of  $y=\left(\left|x\right|\right)$

Part (b). Step 3. Explanation

When $x\ge 0$, We can write that -  $y=f\left(\left|x\right|\right)=f\left(x\right)$

This means that the point to the right of y-axis will remain unchanged.

When $x<0$, We can write that -   $y=f\left(\left|x\right|\right)=f\left(x\right)$

This means that the point to the left of y-axis will be erased and replaced with a reflection of the graph to the right of y-axis.

Part (b). Step 4. Graph

In the graph below - 

The given graph of $y=f\left(x\right)$ is represented by the black solid line.

The new graph of  $y=f\left(\left|x\right|\right)$ is represented by the orange dotted line.

• Point to the right of y-axis will remain unchanged.
• Point to the left of y-axis will be erased and replaced with a reflection of the graph to the right of y-axis.