To determine the transformation that converts the graph of $f(x) = -9(x - 10)^2 - 6$ to the graph of $g(x) = -9(x - 8)^2 - 6$, we will analyze the functions' forms and identify any changes.

Analyzing the Functions

1. Function $f(x) = -9(x - 10)^2 - 6$:

   • Vertex Form: The function is in the vertex form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex.
   • Vertex: The vertex of this parabola is at $(10, -6)$.
   • Direction: The parabola opens downwards because the coefficient of $(x - 10)^2$ is negative.
   • Vertical Stretch: The factor $-9$ indicates a vertical stretch by a factor of 9 and a reflection over the x-axis.

2. Function $g(x) = -9(x - 8)^2 - 6$:

   • Vertex Form: This is also in vertex form, $g(x) = a(x - h)^2 + k$.
   • Vertex: The vertex of this parabola is at $(8, -6)$.
   • Direction: The parabola opens downwards with the same coefficient of $-9$, indicating the same vertical stretch and reflection over the x-axis as in $f(x)$.

Transformation Analysis

Comparing the two functions:

• Horizontal Shift: The vertex of $f(x)$ is at $(10, -6)$, and the vertex of $g(x)$ is at $(8, -6)$. The transformation that occurs here is a shift of 2 units to the right to go from $(10, -6)$ to $(8, -6)$.

• Vertical Shift: There is no vertical shift as the y-coordinate of the vertex ($-6$) remains the same for both functions.

Conclusion

The transformation that converts the graph of $f(x) = -9(x - 10)^2 - 6$ to $g(x) = -9(x - 8)^2 - 6$ is a horizontal shift 2 units to the right.

Would you like more details or have any other questions?

Here are five related questions to expand your understanding:

1. How does a horizontal shift affect the vertex form of a quadratic function?
2. What would the graph of $f(x) = -9(x - 10)^2 - 6$ look like if it was shifted 5 units up?
3. How does changing the coefficient of $(x - h)^2$ affect the graph of a quadratic function?
4. Can you explain the effects of combining horizontal and vertical shifts on a quadratic graph?
5. How would you determine the transformation if $g(x)$ had a different vertical stretch factor than $f(x)$?

Tip: Remember that horizontal shifts are reflected in changes to the $h$ value of the vertex form $(x - h)$, while vertical shifts change the $k$ value.