Short Answer 
The parent function is identified as f(x) = 7^x, and the transformed function is g(x) = 3 * 7^{-x} + 2, which results from reflecting f(x) across the y-axis, applying a vertical stretch by a factor of 3, and shifting the graph upwards by 2 units. The transformed function can be represented as g(x) = 3f(-x) + 2, illustrating the effects of the transformations on the parent function.
Step 1: Identify the Parent Function
The parent function for the given transformation is f(x) = 7^x. This function serves as the baseline from which all modifications will be drawn to create the new function g(x). Recognizing the parent function is crucial because it illustrates how transformations will affect the original graph.
Step 2: Apply Transformations
The new function g(x) = 3 * 7^{-x} + 2 incorporates several transformations of the parent function. These transformations consist of:
- Reflection across the y-axis: The negative sign in the exponent reflects the graph of f(x) across the y-axis.
- Vertical stretch by a factor of 3: The coefficient 3 multiplies the original function’s values, making the graph taller.
- Upward shift by 2 units: The addition of 2 shifts the entire graph upwards, affecting all points on the curve.
Step 3: Write the Transformed Function
After applying the transformations, we can express the new function as g(x) = 3f(-x) + 2. This representation highlights how the parent function f(x) is manipulated. Understanding this relationship clarifies how each transformation contributes to the final function graphically and algebraically.