Let's break this down step by step in a simple way:

Step 1: Choose a parent function

We’ll pick the absolute value function as our parent function. It's simple and easy to understand, and its graph forms a "V" shape.

• Parent function: $f(x) = |x|$

Step 2: Apply transformations

Now, we will apply transformations to this parent function. Here are the transformations in the correct order:

1. Reflection – We will reflect it vertically. This means that the "V" shape will flip upside down, so it will look like an upside-down "V". The new function will be:
   $f(x) = -|x|$

2. Dilation – Next, we will make the graph wider (this is a vertical dilation). We will stretch it by a factor of 2. This means the graph will look like it’s been pulled out vertically, making it flatter. The new function will be: $f(x) = -2|x|$

3. Horizontal translation – Now, we will shift the graph right by 3 units. This will move the "V" shape to the right side of the graph. The new function will be: $f(x) = -2|x - 3|$

4. Vertical translation – Finally, we will shift the graph up by 4 units. This means we move the graph upwards. The final function will be: $f(x) = -2|x - 3| + 4$

Step 3: Graph both the parent function and the transformed function

To help you visualize, I will show the graph of both functions. One will be the original $f(x) = |x|$, and the other will be the transformed function $f(x) = -2|x - 3| + 4$.

Let me create the graph now.

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Step 4: Explanation of the family of functions and transformations

We chose the absolute value function $f(x) = |x|$, which is a member of the piecewise function family. This family is special because the graph has sharp corners (or "V" shapes) rather than smooth curves like other functions (like parabolas).

Here’s what happened in each transformation:

1. Reflection: We reflected the graph vertically, flipping it upside down.
2. Dilation: We stretched it vertically by a factor of 2, which made it wider.
3. Horizontal translation: We moved the graph 3 units to the right, shifting the "V" shape.
4. Vertical translation: We moved the graph 4 units up, shifting the entire graph higher.

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Would you like any more details about any of these transformations? Or do you have other questions?

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Further questions to explore:

1. How do dilations change the width of a graph?
2. What happens if you reflect a function horizontally instead of vertically?
3. How do you tell if a function is translated left or right just by looking at the equation?
4. What are other examples of piecewise functions?
5. How do you find the intercepts of a transformed function?

Tip: Always remember the order of transformations matters! Start with reflections and dilations before translations.