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Let's go over the math problem step by step based on what I can see in the image.

### Question 1: Transformations of \( g(x) = |x| + 4 \)

You are asked to graph the function \( g(x) = |x| + 4 \) as a transformation of the parent function \( f(x) = |x| \). This is a vertical shift, and as written on the sheet, the transformation moves the graph **4 units up**.

- **Domain**: The domain remains the same as the parent function \( f(x) = |x| \), which is all real numbers, or \( (-\infty, \infty) \).
- **Range**: Since the entire graph is shifted up by 4 units, the range starts at 4 and goes to infinity, or \( [4, \infty) \).

### Question 2: Reflection across the x-axis

The problem asks you to reflect the function \( f(x) = x^2 - 4 \) across the x-axis. Reflecting across the x-axis involves multiplying the function by \( -1 \), so the new equation becomes:
\[
f(x) = -(x^2 - 4) = -x^2 + 4.
\]

### Question 3: Reflection across the y-axis

When reflecting a function across the y-axis, you replace \( x \) with \( -x \). So, reflecting \( f(x) = x^2 - 4 \) across the y-axis gives:
\[
f(x) = (-x)^2 - 4 = x^2 - 4.
\]
Since \( x^2 \) is symmetric about the y-axis, the function remains the same.

### Question 4: Vertical Stretch

For \( g(x) = 4.5|x| \), this represents a vertical stretch of the parent function \( f(x) = |x| \) by a factor of 4.5. The V-shape of the absolute value graph becomes steeper due to the vertical stretch.

### Question 5: Vertical Compression

For \( g(x) = 0.5x^2 \), this is a vertical compression of the parent function \( f(x) = x^2 \) by a factor of 0.5, making the parabola wider than the original.

### Question 6: Shift

The function \( g(x) = 3(x + 2)^2 \) represents a parabola shifted 2 units to the left and vertically stretched by a factor of 3.

### Question 7: Shift

For \( g(x) = -(x - 5)^2 + 2 \), the function is shifted 5 units to the right, flipped across the x-axis (because of the negative sign), and shifted 2 units upward.

Would you like further explanation on any of these steps? Here are five related questions to deepen your understanding:

1. How would the graph of \( g(x) = |x| - 3 \) compare to the parent function \( f(x) = |x| \)?
2. What would the range be for the function \( g(x) = -|x| + 2 \)?
3. How does multiplying the function \( f(x) = x^2 \) by a number greater than 1 affect its graph?
4. What happens if you reflect \( f(x) = |x| \) over both the x-axis and the y-axis?
5. How would you graph the function \( g(x) = -|x + 3| \)?

**Tip:** When transforming functions, it's helpful to sketch the parent function first and apply each transformation step by step to ensure accuracy.

Graph the function g(x) = |x| + 4 as a translation of the parent function f(x) shown. How did the transformation affect the domain and range? Reflect across the x-axis: f(x) = -(x^2 - 4). Reflect across the y-axis: f(x) = x^2 - 4. Graph each function as a vertical stretch or compression of the parent function. g(x) = 4.5|x|. g(x) = 0.5x^2.

This page explains how to transform functions through vertical shifts, reflections across the x and y-axes, and vertical stretches/compressions. Learn how to reflect and shift absolute value and quadratic functions with examples.

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