https://file.aiquickdraw.com/mathbot/file/172732965836506lj1at4.jpg

Let's go through the questions in the image step by step:

### 1. Given the graph of the function \( y = f(x) \), sketch the graph of each transformed function:
   - **(a) \( y - 3 = f(x) \)**:
     - This transformation represents a **vertical shift**. Since \( y - 3 = f(x) \), we can rewrite it as \( y = f(x) + 3 \). This means the graph of \( f(x) \) is shifted **up by 3 units**.

   - **(b) \( h(x) = f(x + 1) \)**:
     - This transformation represents a **horizontal shift**. \( f(x + 1) \) shifts the graph **left by 1 unit** (since the transformation is inside the function, it affects the graph in the opposite direction).

   - **(c) \( y + 1 = f(x - 2) \)**:
     - This can be rewritten as \( y = f(x - 2) - 1 \). It involves two transformations:
       - A **horizontal shift right by 2 units** (since \( x - 2 \)).
       - A **vertical shift down by 1 unit** (due to \( -1 \)).

### 2. Describe how to translate the graph of \( y = |x| \) to obtain the graph of the function shown:
   - The graph given is a transformation of \( y = |x| \), a basic "V" shape.
   - It appears to be shifted **right by 4 units** and **down by 2 units**.
   - The equation for this transformation in the form \( y - k = |x - h| \) would be:
     \[
     y + 2 = |x - 4|
     \]
     Or equivalently:
     \[
     y = |x - 4| - 2
     \]

Let me know if you'd like further clarifications or more details on these transformations!

Here are 5 related questions to explore further:
1. How do vertical stretches or compressions affect the graph of a function?
2. How would the graph of \( y = f(x) \) change if the equation were \( y = -f(x) \)?
3. What does \( y = f(-x) \) do to the graph of \( f(x) \)?
4. How do you apply a reflection over the x-axis to a function?
5. How does adding a constant inside the absolute value (e.g., \( |x + 3| \)) affect the graph?

### Tip:
Always check whether the transformation is inside or outside the function to determine whether it affects the graph horizontally or vertically.

Given the graph of the function y=f(x), sketch the graph of each transformed function: a) y-3=f(x), b) h(x)=f(x+1), c) y+1=f(x-2). Describe how to translate the graph of y=|x| to obtain the graph of the function shown, and write the equation of the transformed function in the form y-k=|x-h|.

Learn how to apply vertical and horizontal shifts to a function graph with examples. This problem includes transformations of the graph y=f(x) and absolute value transformations y-k=|x-h|. Suitable for students in Grades 9-10.

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