In the image, the red curve represents the graph of the function $y = \sqrt{x}$, and you are being asked to transform this red graph so that it matches the blue graph.

To perform this transformation, let's analyze the differences between the red and blue graphs:

1. Vertical shift: The blue graph appears to be below the red graph by a certain amount. This indicates a vertical shift downward.

2. Reflection: The blue graph is in the lower half of the coordinate plane, suggesting that there may be a reflection over the x-axis.

Suggested Transformations:

1. Reflection over the x-axis: To reflect the red graph over the x-axis, we multiply the function by $-1$. The new function would be: $y = -\sqrt{x}$ This transformation flips the red graph upside down.

2. Vertical shift: If the blue graph is consistently lower by a certain amount, you would also need to subtract a constant value. Suppose the blue graph is shifted down by 5 units, the transformed equation becomes: $y = -\sqrt{x} - 5$ This shifts the graph downward by 5 units.

You can try applying these transformations step by step to align the red graph with the blue one.

Would you like me to explain more details or make further adjustments?

5 Related Questions:

1. What happens when you apply a horizontal shift to the graph $y = \sqrt{x}$?
2. How do horizontal and vertical stretches affect the graph of $y = \sqrt{x}$?
3. What is the general form for reflecting a graph over the x-axis or y-axis?
4. How do you determine the magnitude of a vertical shift in a graph?
5. What other transformations can be applied to a square root function?

Tip: When transforming functions, try one transformation at a time (e.g., reflection, shifting, stretching) to clearly see the effect of each transformation.