Question
the function f(x)=x^2 was transformed to create the function g(x)=f(x-1)-2. what are the coordinates of the vertex of the graph g?
Answers
GPT-4o mini
To find the coordinates of the vertex of the function \( g(x) = f(x-1) - 2 \), where \( f(x) = x^2 \), we can break down the transformations step by step.
1. **Original Vertex**: The function \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin \( (0, 0) \).
2. **Horizontal Shift**: The transformation \( f(x-1) \) indicates a horizontal shift to the right by 1 unit. This moves the vertex from \( (0, 0) \) to \( (1, 0) \).
3. **Vertical Shift**: The transformation \( -2 \) indicates a downward shift by 2 units. This moves the vertex from \( (1, 0) \) to \( (1, -2) \).
Therefore, the coordinates of the vertex of the graph \( g(x) \) are \( (1, -2) \).
1. **Original Vertex**: The function \( f(x) = x^2 \) is a parabola that opens upwards with its vertex at the origin \( (0, 0) \).
2. **Horizontal Shift**: The transformation \( f(x-1) \) indicates a horizontal shift to the right by 1 unit. This moves the vertex from \( (0, 0) \) to \( (1, 0) \).
3. **Vertical Shift**: The transformation \( -2 \) indicates a downward shift by 2 units. This moves the vertex from \( (1, 0) \) to \( (1, -2) \).
Therefore, the coordinates of the vertex of the graph \( g(x) \) are \( (1, -2) \).