Exercise 1: g(x) = -f(4x) - 8, vertex at (3, -7).
Exercise 2: Many possible transformations, two examples provided.
Exercise 1:
Rule for g:
Horizontal compression by a factor of 1/4: This transformation squeezes the graph horizontally, making it four times wider. Mathematically, we can represent this by multiplying x by 4 in the function: g(x) = f(4x).
Translation 8 units down: This shifts the entire graph 8 units downward. We can achieve this by adding -8 to the function: g(x) = f(4x) - 8.
Reflection across the x-axis: This flips the graph upside down. We can represent this by multiplying the function by -1: g(x) = -f(4x) - 8.
Therefore, the rule for g is: g(x) = -f(4x) - 8.
Vertex of g:
The vertex of the original function f(x) = (x - 12)² + 1 is at (12, 1). When we compress horizontally by a factor of 1/4, the vertex moves to (3, 1). Finally, the translation 8 units down shifts the vertex to (3, -7).
Exercise 2:
There are many possible combinations of transformations that can result in the original function f(x) = (x - 1)² + 1. Here are two examples:
1. Reflection, vertical stretch, and translation:
Reflect the graph across the x-axis: g(x) = -f(x).
Stretch the graph vertically by a factor of 2: g(x) = -2f(x).
Translate the graph 5 units down: g(x) = -2f(x) - 5.
2. Horizontal shift, vertical compression, and reflection:
Shift the graph 3 units to the right: g(x) = f(x - 3).
Compress the graph vertically by a factor of 1/2: g(x) = 1/2f(x - 3).
Reflect the graph across the x-axis: g(x) = -1/2f(x - 3).
These are just two examples, and there are countless other valid combinations of transformations that can achieve the same result. The key is to understand the different types of transformations available and how they affect the graph.