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.