To transform the function $y = x^2$ according to the specified operations—reflection about the x-axis, horizontal stretch, and horizontal shift—the steps are as follows:

1. Reflection about the x-axis: Reflecting the function $y = x^2$ about the x-axis results in $y = -x^2$.

2. Stretch horizontally by a factor of 5: To stretch horizontally by a factor of 5, replace $x$ with $\frac{x}{5}$. This gives $y = -\left(\frac{x}{5}\right)^2$.

3. Shift 62 units to the right: To shift the function 62 units to the right, replace $x$ with $x - 62$. The function then becomes $y = -\left(\frac{x - 62}{5}\right)^2$.

So, the transformed function is:

$y = -\left(\frac{x - 62}{5}\right)^2$

Would you like further details or have any questions?

Here are some related questions to deepen your understanding:

1. How would the function change if it were shifted 62 units to the left instead of to the right?
2. What happens to the function if it is reflected about the y-axis instead of the x-axis?
3. How would the transformation differ if the function were stretched vertically by a factor of 5 instead of horizontally?
4. What is the effect of reflecting a function about both the x-axis and y-axis?
5. How does the graph of the original function compare to the transformed function in terms of its vertex?

Tip: When performing transformations, always apply them in the following order: reflections, stretches/compressions, and then shifts.