Step 1: Identify the Parent Function

The function described involves both linear and non-negative values of $y$, with sharp turns and a V-like shape. Based on this information, the parent function is most likely:

$y = |x|$

This is because the absolute value function, $y = |x|$, has a V-shape and is always non-negative, fitting the description of the graph entering the second quadrant, changing direction, and having linear segments on either side of the vertex.

Step 2: Identify the Transformation Represented by the Graph

Now let's identify the transformations based on the description of the graph. Here's a step-by-step breakdown of the changes:

• Vertical shift of 3 units upward: The graph crosses the $y$-axis at $y = 3$, whereas the parent function $y = |x|$ crosses the $y$-axis at $y = 0$. This indicates a vertical shift of 3 units upward.

• Reflection in the x-axis: There's no mention of the graph being reflected over the x-axis (it stays above the x-axis), so this does not apply.

• Horizontal shift of 3 units to the right: The function changes direction at $x = -3$, which corresponds to a shift of 3 units to the right from the parent function's vertex at $x = 0$. This suggests a horizontal shift of 3 units to the right.

• Vertical stretch: There's no indication of a stretching in the vertical direction (the function seems to maintain its "V" shape), so this does not apply.

• Horizontal shift of 3 units to the left: The vertex of the transformed function appears at $x = -3$, not at $x = 0$, indicating a horizontal shift to the left of 3 units (not right, as we first suspected).

• Vertical shift of 3 units downward: The graph is shifted upward by 3 units, not downward. Therefore, this does not apply.

Summary of Transformations:

• Vertical shift of 3 units upward
• Horizontal shift of 3 units to the left

Final Equation

Given the transformations, the equation of the transformed function is:

$y = |x + 3| + 3$

This reflects:

• The horizontal shift of 3 units to the left (hence the $+3$ inside the absolute value),
• The vertical shift of 3 units upward (the $+3$ outside the absolute value).

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Would you like to explore more details about transformations or the graphing process? Here are 5 related questions for further learning:

1. How does the graph of a function change when a horizontal shift and vertical shift occur together?
2. What would happen to the graph if the function were reflected over the x-axis?
3. How do you identify the effects of transformations on other functions like $y = x^2$ or $y = \sin(x)$?
4. Can you explain how transformations affect the domain and range of the absolute value function?
5. How do you graph $y = |x - 2| + 5$ step by step?

Tip: When dealing with absolute value transformations, always pay attention to the shifts inside and outside the absolute value, as they affect horizontal and vertical positioning, respectively!