We are given the function $y = x^2$ and asked to apply several transformations to it:

1. Horizontally stretch the graph by a factor of 2.
2. Reflect the graph in the x-axis.
3. Shift the graph 3 units down.

Step-by-step transformation process:

Step 1: Horizontal Stretch by a factor of 2

A horizontal stretch by a factor of 2 means we replace $x$ with $\frac{x}{2}$. The transformation for horizontal stretching is:

$y = \left( \frac{x}{2} \right)^2 = \frac{x^2}{4}$

So, after this transformation, the equation becomes: $y = \frac{x^2}{4}$

Step 2: Reflection in the x-axis

A reflection in the x-axis means we multiply the entire function by $-1$. Applying this to $y = \frac{x^2}{4}$:

$y = -\frac{x^2}{4}$

Step 3: Shift 3 units down

To shift the graph 3 units down, we subtract 3 from the function. Applying this to $y = -\frac{x^2}{4}$:

$y = -\frac{x^2}{4} - 3$

Final Equation:

The final transformed equation is:

$y = -\frac{x^2}{4} - 3$

Graphing the Transformed Equation:

To graph $y = -\frac{x^2}{4} - 3$:

1. Shape: This is a parabola that opens downward (due to the negative sign).
2. Stretching: The parabola is wider than the basic $y = x^2$ because of the horizontal stretch by a factor of 2.
3. Shift: The vertex is located at $(0, -3)$, since the graph has been shifted 3 units downward.
4. Reflection: The graph is upside-down because of the reflection across the x-axis.

Would you like me to graph this for you?

Follow-up Questions:

1. How do horizontal stretches affect the width of a parabola?
2. What does reflecting a function in the x-axis do to its graph?
3. How does shifting a graph vertically influence the equation?
4. Can you explain how to determine the vertex of the transformed parabola?
5. What would the equation be if instead of shifting down, we shifted the graph upwards by 3 units?

Tip:

Always apply transformations in the correct order: stretch/shrink first, then reflect, and finally shift.