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Let's analyze the problem step by step.

You are given that the point \((-3, -6)\) lies on the graph of \( y = f(x) \), meaning \( f(-3) = -6 \). Now, we need to find points on different transformations of the function \( f(x) \), which are represented by \( g(x) \) in each case. Let's go through each subpart:

### Part (a):
You are asked to find a point on the graph of \( y = g(x) \), where \( g(x) = f(x) + 3 \).

Since \( g(x) = f(x) + 3 \), it means we add 3 to the value of \( f(x) \). For the point \( (-3, -6) \), we know \( f(-3) = -6 \), so:

\[
g(-3) = f(-3) + 3 = -6 + 3 = -3
\]

Thus, the point on the graph of \( g(x) \) is \( (-3, -3) \).

### Part (b):
Here, \( g(x) = 2f(x) \).

We need to multiply the value of \( f(x) \) by 2. For \( f(-3) = -6 \):

\[
g(-3) = 2 \cdot f(-3) = 2 \cdot (-6) = -12
\]

Thus, the point on the graph of \( g(x) \) is \( (-3, -12) \).

### Part (c):
In this part, \( g(x) = \frac{1}{6} f(x) \).

Here, we multiply the value of \( f(x) \) by \( \frac{1}{6} \). Since \( f(-3) = -6 \):

\[
g(-3) = \frac{1}{6} \cdot f(-3) = \frac{1}{6} \cdot (-6) = -1
\]

Thus, the point on the graph of \( g(x) \) is \( (-3, -1) \).

### Part (d):
In this part, \( g(x) = -f(x) \).

We need to take the negative of the value of \( f(x) \). For \( f(-3) = -6 \):

\[
g(-3) = -f(-3) = -(-6) = 6
\]

Thus, the point on the graph of \( g(x) \) is \( (-3, 6) \).

### Part (e):
Here, \( g(x) = f(-x) \).

This reflects the function across the y-axis. So, instead of \( f(-3) \), we need to evaluate \( f(3) \). However, since we do not know the value of \( f(3) \), we cannot determine the exact answer without additional information.

### Part (f):
In this part, \( g(x) = f(x + 8) \).

This means we need to evaluate \( f \) at \( -3 + 8 = 5 \), which requires the value of \( f(5) \). Again, without knowing \( f(5) \), we cannot determine the exact point.

---

Would you like more details on any of these transformations or further clarifications? Here are 5 related questions to deepen understanding:

1. What happens to a function when it is shifted vertically by adding or subtracting a constant?
2. How does multiplying a function by a constant affect its graph?
3. What is the effect of reflecting a function across the y-axis on its graph?
4. How do horizontal shifts (e.g., \( f(x + c) \)) affect the graph of a function?
5. How can we determine if two functions are inverses based on their graphs?

**Tip**: Understanding how transformations (shifts, stretches, reflections) affect the graph of a function is key in function analysis and graphing.

The point (-3, -6) is on the graph of y = f(x). Find the points on the graph of y = g(x) for different transformations of f(x) as given below.

Find out how the point (-3, -6) on the graph of y = f(x) is transformed under various transformations such as vertical shifts, vertical stretches, reflections, and horizontal shifts. This guide explains the step-by-step process for each transformation, ideal for high school algebra students.

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