Question
Which of the following best describes the transformation that occurs when changing an equation from f(x)=x2+1
to f(x)=(x−3)2+1
?(1 point)
Responses
The graphed function would shift 3 units to the left but otherwise have the same shape.
The graphed function would shift 3 units to the left but otherwise have the same shape.
The graphed function would shift 3 units to the right but otherwise have the same shape.
The graphed function would shift 3 units to the right but otherwise have the same shape.
The graphed function would shift 3 units to the right and would be half as wide.
The graphed function would shift 3 units to the right and would be half as wide.
The graphed function would shift 3 units to the left and would be half as wide.
The graphed function would shift 3 units to the left and would be half as wide.
to f(x)=(x−3)2+1
?(1 point)
Responses
The graphed function would shift 3 units to the left but otherwise have the same shape.
The graphed function would shift 3 units to the left but otherwise have the same shape.
The graphed function would shift 3 units to the right but otherwise have the same shape.
The graphed function would shift 3 units to the right but otherwise have the same shape.
The graphed function would shift 3 units to the right and would be half as wide.
The graphed function would shift 3 units to the right and would be half as wide.
The graphed function would shift 3 units to the left and would be half as wide.
The graphed function would shift 3 units to the left and would be half as wide.
Answers
GPT-4o mini
To analyze the transformation from the equation \( f(x) = x^2 + 1 \) to \( f(x) = (x - 3)^2 + 1 \), we can break it down as follows:
1. The term \( (x - 3) \) indicates a horizontal shift. Specifically, if a function is transformed from \( f(x) \) to \( f(x - h) \), it shifts \( h \) units to the right. Therefore, \( (x - 3) \) represents a shift of 3 units to the right.
2. The \( +1 \) at the end of both equations indicates that both functions are 1 unit up vertically, which remains unchanged in both transformations.
3. The shape of the function, a parabola in this case, does not change width since there is no coefficient affecting the \( x^2 \) term in front of the squared term.
Given these observations, the correct response is:
**The graphed function would shift 3 units to the right but otherwise have the same shape.**
1. The term \( (x - 3) \) indicates a horizontal shift. Specifically, if a function is transformed from \( f(x) \) to \( f(x - h) \), it shifts \( h \) units to the right. Therefore, \( (x - 3) \) represents a shift of 3 units to the right.
2. The \( +1 \) at the end of both equations indicates that both functions are 1 unit up vertically, which remains unchanged in both transformations.
3. The shape of the function, a parabola in this case, does not change width since there is no coefficient affecting the \( x^2 \) term in front of the squared term.
Given these observations, the correct response is:
**The graphed function would shift 3 units to the right but otherwise have the same shape.**