The original function is f(x) = x.
Three transformations are applied: horizontal compression by a factor of 1/4, translation 8 units down, and reflection across the x-axis.
The resulting function rule for g(x) is: g(x) = - (1/16) * x^2 + 8.
Here's the complete calculation for the function rule of g(x) based on the given graph and transformations:
Original Function:
f(x) = x
Transformation 1: Horizontal Compression by a Factor of 1/4:
This transformation shrinks the graph horizontally by a factor of 4. Mathematically, we can represent this by multiplying x by 1/4: g(x) = f(1/4 * x) = (1/4 * x)^2 = (1/16) * x^2
Transformation 2: Translation 8 Units Down:
This transformation shifts the entire graph 8 units downward. We can achieve this by subtracting 8 from the function: g(x) = (1/16) * x^2 - 8
Transformation 3: Reflection across the x-axis:
This transformation flips the graph upside down. We can represent this by multiplying the function by -1: g(x) = -[(1/16) * x^2 - 8] = -(1/16) * x^2 + 8
Therefore, the complete function rule for g(x) after all the transformations is:
g(x) = - (1/16) * x^2 + 8