Write the function,
y
=
|
x
|
after the following transformations:

1) Reflect the function about the y-axis.
2) Compress the function horizontally by a factor of 1/2.
3) Shift the function left 84 units.

y
=

Let's work through the transformations step by step on the absolute value function \( y = |x| \).

### 1. Reflect the function about the y-axis:
Reflecting \( y = |x| \) about the y-axis results in \( y = |-x| \). However, because the absolute value function is symmetrical around the y-axis, reflecting it about the y-axis does not change it. So, the function remains \( y = |x| \).

### 2. Compress the function horizontally by a factor of 1/2:
To compress the function horizontally by a factor of 1/2, we replace \( x \) with \( 2x \). The function becomes:
\[
y = |2x|
\]

### 3. Shift the function left 84 units:
To shift the function left by 84 units, we replace \( x \) with \( x + 84 \) inside the absolute value. The function now becomes:
\[
y = |2(x + 84)|
\]

This can be simplified as:
\[
y = |2x + 168|
\]

So the final transformed function is:
\[
\boxed{y = |2x + 168|}
\]

Would you like a detailed explanation or have any questions about these transformations? Here are some related questions:

1. How would the function change if we shifted it right instead of left?
2. What happens if we stretch the function vertically instead of horizontally?
3. How would the function change if we reflected it about the x-axis instead of the y-axis?
4. Can you apply a vertical translation to the final function? What would it look like?
5. How would you describe the effect of each transformation on the graph of the original function?

**Tip:** When applying multiple transformations to a function, follow the order: reflections, dilations (stretches/compressions), and then translations (shifts).

Explore how to transform the absolute value function y = |x| through reflection about the y-axis, horizontal compression by a factor of 1/2, and shifting left by 84 units. Detailed step-by-step explanation provided.

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Write the function, y

y

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