Sure, let's break this down step-by-step.
### Part a) Reflection of the Graph [tex]\( y = x^2 \)[/tex]
1. Identify the initial minimum point:
The minimum point of the graph [tex]\( y = x^2 \)[/tex] is at [tex]\((0, 0)\)[/tex].
2. Transformation Description:
Reflecting the graph [tex]\( y = x^2 \)[/tex] in the line [tex]\( y = 1 \)[/tex].
3. Calculate the new equation after reflection:
When the graph [tex]\( y = x^2 \)[/tex] is reflected in the line [tex]\( y = 1 \)[/tex], each point [tex]\( (x, y) \)[/tex] on the original graph is transformed such that the new [tex]\( y \)[/tex]-coordinate is [tex]\( y = 2 - y' \)[/tex].
Therefore, the equation [tex]\( y = x^2 \)[/tex] transforms to:
[tex]\( y = -x^2 + 2 \)[/tex].
4. Determine the coordinates of the new minimum point:
Originally, the minimum point of [tex]\( y = x^2 \)[/tex] is at [tex]\( (0, 0) \)[/tex]. After reflecting in [tex]\( y = 1 \)[/tex], the new coordinates of the minimum point will be [tex]\( (0, 1) \)[/tex].
Answer:
- Equation after reflection: [tex]\( y = -x^2 + 2 \)[/tex]
- New minimum point: [tex]\( (0, 1) \)[/tex]
### Part b) Shifting the Graph [tex]\( y = x^2 \)[/tex] Down
1. Description of the transformation:
The graph [tex]\( y = x^2 \)[/tex] is shifted down by 3 units.
2. Calculate the new equation after transformation:
Moving the graph [tex]\( y = x^2 \)[/tex] down by 3 units, each [tex]\( y \)[/tex]-coordinate will decrease by 3.
Thus, the equation becomes:
[tex]\( y = x^2 - 3 \)[/tex].
Answer:
- Equation after moving down: [tex]\( y = x^2 - 3 \)[/tex]
### Part c) Shift and Transformation
1. Description of the transformation:
The graph [tex]\( y = x^2 \)[/tex] is transformed to give [tex]\( y = (x - 2)^2 \)[/tex].
2. Transformation Description:
Shifting the graph [tex]\( y = x^2 \)[/tex] to the right by 2 units.
3. Determine the new minimum point:
The original minimum point of [tex]\( y = x^2 \)[/tex] is at [tex]\( (0, 0) \)[/tex]. After shifting to the right by 2 units, the new coordinates of the minimum point will be [tex]\( (2, 0) \)[/tex].
Answer:
- Single transformation: A horizontal shift to the right by 2 units.
- New minimum point: [tex]\( (2, 0) \)[/tex]
Summarizing:
1. Reflection of [tex]\( y = x^2 \)[/tex] in [tex]\( y = 1 \)[/tex]:
- New Equation: [tex]\( y = -x^2 + 2 \)[/tex]
- Minimum Point: [tex]\( (0, 1) \)[/tex]
2. Shift [tex]\( y = x^2 \)[/tex] down by 3 units:
- New Equation: [tex]\( y = x^2 - 3 \)[/tex]
3. Transform [tex]\( y = x^2 \)[/tex] to [tex]\( y = (x - 2)^2 \)[/tex]:
- Transformation: A right shift by 2 units.
- New minimum point: [tex]\( (2, 0) \)[/tex]
