Given that f(x) = x^2-7, the graph of function h can be formed by transforming f as follows:, horizontally stretching f by a factor of 3, vertically reflecting the result across the horizontal axis. What is the formula for h?

To find the formula for the function \( h(x) \), we start with the function \( f(x) = x^2 - 7 \) and apply the given transformations:

1. **Horizontally stretching \( f(x) \) by a factor of 3**:
   - To horizontally stretch a function by a factor of \( k \), replace \( x \) with \( \frac{x}{k} \) in the function.
   - Therefore, the function after the horizontal stretch by a factor of 3 is \( f\left(\frac{x}{3}\right) = \left(\frac{x}{3}\right)^2 - 7 \).

2. **Vertically reflecting the result across the horizontal axis**:
   - To reflect a function vertically (i.e., across the x-axis), multiply the entire function by \( -1 \).
   - So, after reflecting, the function becomes \( h(x) = -\left(\left(\frac{x}{3}\right)^2 - 7\right) \).

Now, simplify the expression for \( h(x) \):

\[
h(x) = -\left(\frac{x^2}{9} - 7\right)
\]

Distribute the negative sign:

\[
h(x) = -\frac{x^2}{9} + 7
\]

Thus, the formula for the function \( h(x) \) is:

\[
h(x) = -\frac{x^2}{9} + 7
\]

Would you like more details or have any questions? Here are some related questions to consider:

1. What happens if you horizontally compress \( f(x) \) instead of stretching it?
2. How does the vertex of the parabola change after the transformation?
3. What would the formula look like if we reflected \( f(x) \) across the y-axis instead?
4. How does the graph of \( h(x) \) compare to the original graph of \( f(x) \)?
5. Can you describe the range of the function \( h(x) = -\frac{x^2}{9} + 7 \)?

**Tip:** When performing multiple transformations, it's crucial to follow the correct order, especially when mixing horizontal and vertical transformations.

Explore how to transform the quadratic function f(x) = x^2 - 7 by horizontally stretching it by a factor of 3 and vertically reflecting it across the x-axis. Detailed step-by-step solution with formulas and examples provided.

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