6.

Question 1

To write the function rule for \( g(x) \), we start from \( f(x) = x^2 \).

Assuming \( g(x) \) is a translation of \( f(x) \) in the form \( g(x) = a(x - h)^2 + k \), we need to identify the values of \( a \), \( h \), and \( k \).

1. Identify Translation:

   • Determine the value of \( h \) and \( k \) by looking at how the graph of \( f(x) \) is transformed to get \( g(x) \).
   • If the graph shifts to the left or right, \( h \) will be negative or positive respectively.
   • If the graph shifts up or down, \( k \) will be positive or negative respectively.

2. Find the Point for Scaling \( a \):

   • Use another point on the graph of \( g(x) \) (besides any provided) to determine \( a \).
   • Substitute the coordinates of that point into the equation \( g(x) = a(x - h)^2 + k \) and solve for \( a \).

Without explicit values from a graph, I can't provide the exact function for \( g(x) \). If you provide specific points or transformations observed, I can assist with the calculations.

Question 2

The original function has the point (10, 32) and translates to (5, 32).

1. Change in X-Coordinate:

   • \( 10 \) to \( 5 \) is a shift to the left by \( 5 \) units.

2. Change in Y-Coordinate:

   • The y-coordinate remains constant at \( 32 \), meaning the slope and vertical position of points on the graph do not change, confirming that the slope remains the same.

Conclusion:

The correct description of the translation is:

The function is translated 5 units to the left and has the same slope.