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.