The graph of is horizontally stretched
by a factor of 2, reflected in the x-axis, and
shifted 3 units down. Find the equation that
results from the transformation, and graph it

To find the equation resulting from the given transformations, we need to apply the transformations step-by-step to a generic function \( f(x) \). Let's assume we start with a basic function \( y = f(x) \).

### Step-by-Step Transformation

1. **Horizontal Stretch by a Factor of 2:**
   A horizontal stretch by a factor of 2 changes the function from \( f(x) \) to \( f\left(\frac{x}{2}\right) \). 

2. **Reflection in the x-axis:**
   Reflecting a function in the x-axis changes it to \( -f(x) \). So, our function now becomes:
   \[
   -f\left(\frac{x}{2}\right)
   \]

3. **Shift 3 Units Down:**
   To shift the function 3 units down, subtract 3 from the function. The final transformed function becomes:
   \[
   y = -f\left(\frac{x}{2}\right) - 3
   \]

### Example Transformation

If we start with a common function, like \( f(x) = x^2 \):

1. Apply the horizontal stretch:
   \[
   f\left(\frac{x}{2}\right) = \left(\frac{x}{2}\right)^2 = \frac{x^2}{4}
   \]

2. Apply the reflection in the x-axis:
   \[
   -f\left(\frac{x}{2}\right) = -\frac{x^2}{4}
   \]

3. Shift 3 units down:
   \[
   y = -\frac{x^2}{4} - 3
   \]

### Resulting Equation
The equation after applying all the transformations is:
\[
y = -\frac{x^2}{4} - 3
\]

### Graphing the Transformed Function

To graph this function:

1. **Start with the basic graph** of \( y = x^2 \).
2. **Stretch it horizontally** by a factor of 2 to get \( y = \frac{x^2}{4} \).
3. **Reflect it over the x-axis** to get \( y = -\frac{x^2}{4} \).
4. **Shift the graph 3 units down** to get \( y = -\frac{x^2}{4} - 3 \).

Let's visualize this by plotting the graph.

I'll create a graph for the original function \( y = x^2 \) and its transformation \( y = -\frac{x^2}{4} - 3 \).It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can still graph the function manually using graphing software or a calculator. The key points to consider are:

1. The original function is \( y = x^2 \).
2. After applying the transformations, the function becomes \( y = -\frac{x^2}{4} - 3 \).
3. Start with the graph of \( y = x^2 \), stretch it horizontally, reflect it across the x-axis, and then shift it down by 3 units.

If you have any questions or want more details, feel free to ask!

### Related Questions
1. How do horizontal stretches and compressions affect the shape of a graph?
2. What are the effects of reflecting a function across different axes?
3. How do vertical and horizontal shifts impact the graph of a function?
4. What are some common transformations of quadratic functions?
5. How would the transformations differ if the function was \( y = \sin(x) \) instead?

**Tip:** Always apply transformations in the order they are given to avoid confusion and mistakes.

The graph of is horizontally stretched by a factor of 2, reflected in the x-axis, and shifted 3 units down. Find the equation that results from the transformation, and graph it.

Discover how to apply horizontal stretches, reflections, and downward shifts to a quadratic function. This guide provides a step-by-step solution to transforming the equation y = x^2 into y = -x^2/4 - 3, suitable for students in Grades 9-11.

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