https://file.aiquickdraw.com/mathbot/file/1726170313986pib9vgcy.jpg

Let's go over the tasks step-by-step based on the image you've uploaded:

### Task 1:
You are given a function \( y = f(x) \) as shown in the graph in Figure 2, and you need to plot the graphs of:
1. \( y = -f(x) \): This reflects the graph over the x-axis (invert the y-values).
2. \( y = f(-x) \): This reflects the graph over the y-axis (invert the x-values).

### Task 2:
You need to plot the following functions:
1. \( y = -3x^2 \): This is a downward-opening parabola since the coefficient of \( x^2 \) is negative.
2. \( y = \sqrt{5x} \): This is a square root function, starting from \( x = 0 \).

### Task 3:
You are asked to find the vertex coordinates of the following parabolas:
1. \( y = x^2 + 7 \): The vertex is at \( (0, 7) \), since the parabola is of the form \( y = x^2 + c \).
2. \( y = (x + 8)^2 \): The vertex is at \( (-8, 0) \), since the form is \( (x - h)^2 \), and in this case, \( h = -8 \).
3. \( y = (x - 6)^2 + 9 \): The vertex is at \( (6, 9) \).

### Task 4:
You are asked to plot the graph of \( y = \sqrt{x} \), and based on that:
1. \( y = \sqrt{x - 4} \): This is a horizontal shift of the square root function to the right by 4 units.
2. \( y = 3 + \sqrt{x + 1} \): This is a vertical shift up by 3 units and a horizontal shift to the left by 1 unit.

Additionally, you are asked to plot \( y = \frac{2x}{x + 3} \), which is a rational function.

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Would you like details on how to reflect or shift the graphs, or any specific part explained further?

Here are 5 related questions to expand on these topics:
1. How do we determine the range of the function \( y = \sqrt{5x} \)?
2. What is the effect of shifting \( y = \sqrt{x} \) by 4 units to the left on its domain and range?
3. How would the graph of \( y = -f(-x) \) differ from the reflections you did earlier?
4. Can you describe the asymptotic behavior of the rational function \( y = \frac{2x}{x + 3} \)?
5. How do the transformations affect the domain and range of the function \( y = -3x^2 \)?

**Tip:** For reflections, it's useful to apply transformations one at a time to avoid mistakes.

На рисунке 2 изображён график функции y = f(x). Постройте график функции: 1) y = -f(x); 2) y = f(-x).

This problem explores graph transformations, including reflections and shifts, based on the graph of the function y = f(x). You will also graph parabolas and square root functions, suitable for students in grades 8-10.

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