Let's analyze the transformation step-by-step given the function [tex]\( g(x) = 3x^2 - 3 \)[/tex], with respect to the parent function [tex]\( y = x^2 \)[/tex].
1. Analyze the coefficient of [tex]\( x^2 \)[/tex]:
- In the parent function [tex]\( y = x^2 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is 1.
- In the function [tex]\( g(x) = 3x^2 - 3 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is 3.
This indicates a vertical stretch by a factor of 3. A coefficient greater than 1 stretches the graph vertically.
2. Analyze the constant term:
- In the parent function [tex]\( y = x^2 \)[/tex], there is no constant term, which is equivalent to having a constant term of 0.
- In the function [tex]\( g(x) = 3x^2 - 3 \)[/tex], there is a constant term of [tex]\(-3\)[/tex].
This indicates a vertical shift downward by 3 units. A negative constant term shifts the graph downwards.
3. Other potential transformations:
- Reflection across the [tex]\(x\)[/tex]-axis: This would occur if the coefficient of [tex]\( x^2 \)[/tex] were negative, which it is not.
- Vertical compression: This would occur if the coefficient of [tex]\( x^2 \)[/tex] were between 0 and 1, which it is not.
- Horizontal shifts: These occur when there is a change to the [tex]\( x \)[/tex]-term inside the squared term, such as [tex]\((x - h)^2\)[/tex], which there is not in this function.
Therefore, we can identify the transformations that have occurred:
- Vertical stretch by a factor of 3.
- Vertical shift downwards by 3 units.
So, the correct identifications are:
1. Vertical stretch
2. Vertical shift of 3 down
