Question
the function f(x)=x^2 was transformed to create the function g(x)=f(x-3)-4. what are the coordinates of the vertex of the graph g?
Answers
GPT-4o mini
The function \( f(x) = x^2 \) has its vertex at the point \( (0, 0) \).
The transformation to create \( g(x) = f(x-3) - 4 \) involves two steps:
1. **Horizontal Shift**: The term \( (x - 3) \) indicates a shift to the right by 3 units. Therefore, the new vertex after this shift becomes \( (3, 0) \).
2. **Vertical Shift**: The term \( -4 \) indicates a shift downward by 4 units. Hence, the y-coordinate of the vertex after this shift becomes \( 0 - 4 = -4 \).
Combining these shifts, the vertex of \( g(x) \) is located at:
\[
(3, -4)
\]
Thus, the coordinates of the vertex of the graph \( g \) are \( \boxed{(3, -4)} \).
The transformation to create \( g(x) = f(x-3) - 4 \) involves two steps:
1. **Horizontal Shift**: The term \( (x - 3) \) indicates a shift to the right by 3 units. Therefore, the new vertex after this shift becomes \( (3, 0) \).
2. **Vertical Shift**: The term \( -4 \) indicates a shift downward by 4 units. Hence, the y-coordinate of the vertex after this shift becomes \( 0 - 4 = -4 \).
Combining these shifts, the vertex of \( g(x) \) is located at:
\[
(3, -4)
\]
Thus, the coordinates of the vertex of the graph \( g \) are \( \boxed{(3, -4)} \).