Find the function that is finally graphed after the following transformations are applied to the graph of y equals StartRoot x EndRoot in the order listed.
​(1)  Reflect about the x dash axis
​(2)  Shift up 3 units
​(3)  Shift right 5 units

We start with the base function:

\[
y = \sqrt{x}
\]

Now, let's apply the transformations step by step:

### 1. Reflect about the x-axis:
To reflect the graph about the x-axis, we multiply the entire function by \(-1\). The function becomes:

\[
y = -\sqrt{x}
\]

### 2. Shift up 3 units:
To shift the graph up by 3 units, we add 3 to the entire function. The function now becomes:

\[
y = -\sqrt{x} + 3
\]

### 3. Shift right 5 units:
To shift the graph to the right by 5 units, we replace \(x\) with \((x - 5)\). The final function is:

\[
y = -\sqrt{x - 5} + 3
\]

Thus, the function that is finally graphed is:

\[
y = -\sqrt{x - 5} + 3
\]

Do you need more details or have any questions?

Here are some related questions you might find interesting:
1. What would the function look like if it were shifted left 5 units instead of right?
2. How would the function change if reflected over the y-axis instead of the x-axis?
3. How does shifting the graph down by 3 units affect the equation?
4. What would the graph look like if no reflection was applied, but the shifts remained the same?
5. How do transformations like stretches or compressions affect square root functions?

**Tip**: Always apply horizontal transformations (shifts left/right) before vertical ones when solving transformation problems.

Find the function that is finally graphed after the following transformations are applied to the graph of y equals StartRoot x EndRoot in the order listed.
(1) Reflect about the x-axis
(2) Shift up 3 units
(3) Shift right 5 units

Explore how to transform the graph of the square root function through reflection, vertical shift, and horizontal shift. The problem demonstrates key function transformations, including reflection across the x-axis, shifting up by 3 units, and moving right by 5 units. Suitable for students in Grades 9-11.

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