To graph each transformed function, let's break down the transformations one by one for each function.

1. $f(x) = -2|x + 1| - 5$

This is an absolute value function with the following transformations:

• Horizontal shift: $x + 1$ means a shift 1 unit to the left.
• Vertical shift: The $-5$ moves the graph 5 units down.
• Vertical stretch: The $-2$ multiplies the absolute value, flipping the graph upside down and stretching it vertically by a factor of 2.

Vertex: $(-1, -5)$

2. $g(x) = (x - 1)^2 - 2$

This is a quadratic function with:

• Horizontal shift: $x - 1$ means a shift 1 unit to the right.
• Vertical shift: The $-2$ moves the graph 2 units down.

Vertex: $(1, -2)$

3. $h(x) = \sqrt{-(x - 6)}$

This is a square root function, which has been reflected and shifted:

• Reflection: The negative sign inside the square root reflects the graph across the y-axis.
• Horizontal shift: $x - 6$ moves the graph 6 units to the right.

Starting point: $(6, 0)$

4. $m(x) = \sqrt{-x - 6}$

This is also a square root function, with different transformations:

• Reflection: The $-x$ reflects it across the y-axis.
• Horizontal shift: The $-6$ moves the graph 6 units left.

Starting point: $(-6, 0)$

5. $n(x) = \frac{1}{3}(x + 3)^2 + 4$

This is a quadratic function that has been compressed and shifted:

• Horizontal shift: $x + 3$ moves the graph 3 units to the left.
• Vertical shift: The $+4$ moves the graph 4 units up.
• Vertical compression: The $\frac{1}{3}$ compresses the graph vertically.

Vertex: $(-3, 4)$

6. $p(x) = 4\sqrt{x + 5} - 1$

This is a square root function that has been stretched and shifted:

• Horizontal shift: $x + 5$ moves the graph 5 units to the left.
• Vertical shift: The $-1$ moves the graph 1 unit down.
• Vertical stretch: The 4 stretches the graph vertically.

Starting point: $(-5, -1)$

7. $r(x) = \frac{3}{4}|x - 5| + 4$

This is another absolute value function:

• Horizontal shift: $x - 5$ moves the graph 5 units to the right.
• Vertical shift: The $+4$ moves the graph 4 units up.
• Vertical compression: The $\frac{3}{4}$ compresses the graph vertically.

Vertex: $(5, 4)$

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Would you like me to show the graphs or provide more details on any specific function?

Here are some related questions to explore:

1. How does a horizontal shift affect the shape of a function's graph?
2. What impact does a vertical stretch or compression have on the graph of a function?
3. How do absolute value transformations affect the slope of the graph?
4. How would you graph a reflection across the x-axis for any given function?
5. What is the domain and range for square root functions after transformations?

Tip: When graphing transformed functions, always start with the basic shape of the function (like $|x|$, $x^2$, or $\sqrt{x}$) and apply each transformation step-by-step.