Answer :
To describe the transformation from the parent quadratic function [tex]\( y = x^2 \)[/tex] to the function [tex]\( y = \frac{1}{4}x^2 \)[/tex], we need to compare the two functions and identify how the changes in coefficients affect the shape of the graph.
### Step-by-Step Description:
1. Identify the Parent Function:
- The parent function is [tex]\( y = x^2 \)[/tex]. This function represents a basic parabolic shape that opens upwards, with its vertex at the origin (0, 0).
2. Identify the Given Function:
- The given function is [tex]\( y = \frac{1}{4}x^2 \)[/tex].
3. Compare the Coefficients:
- In the parent function [tex]\( y = x^2 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is 1.
- In the given function [tex]\( y = \frac{1}{4}x^2 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
4. Determine the Effect of the Coefficient:
- The coefficient in front of [tex]\( x^2 \)[/tex] in a quadratic function indicates how the graph is stretched or shrunk vertically.
- A coefficient greater than 1 would mean a vertical stretch.
- A coefficient between 0 and 1 means a vertical shrink.
5. Conclude the Transformation:
- Since the coefficient in the given function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is [tex]\(\frac{1}{4}\)[/tex], which is between 0 and 1, the transformation is a vertical shrink.
- Specifically, the graph of the function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is a vertical shrink by a factor of [tex]\(\frac{1}{4}\)[/tex]. This means that every y-value of the parent function is scaled down to [tex]\(\frac{1}{4}\)[/tex] of its original value.
### Final Answer:
The transformation from the parent quadratic function [tex]\( y = x^2 \)[/tex] to the function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is a vertical shrink by a factor of [tex]\(\frac{1}{4}\)[/tex]. This reduces the height of each point on the parabola by 75%, resulting in a wider and shorter parabola compared to the original.
### Step-by-Step Description:
1. Identify the Parent Function:
- The parent function is [tex]\( y = x^2 \)[/tex]. This function represents a basic parabolic shape that opens upwards, with its vertex at the origin (0, 0).
2. Identify the Given Function:
- The given function is [tex]\( y = \frac{1}{4}x^2 \)[/tex].
3. Compare the Coefficients:
- In the parent function [tex]\( y = x^2 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is 1.
- In the given function [tex]\( y = \frac{1}{4}x^2 \)[/tex], the coefficient of [tex]\( x^2 \)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
4. Determine the Effect of the Coefficient:
- The coefficient in front of [tex]\( x^2 \)[/tex] in a quadratic function indicates how the graph is stretched or shrunk vertically.
- A coefficient greater than 1 would mean a vertical stretch.
- A coefficient between 0 and 1 means a vertical shrink.
5. Conclude the Transformation:
- Since the coefficient in the given function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is [tex]\(\frac{1}{4}\)[/tex], which is between 0 and 1, the transformation is a vertical shrink.
- Specifically, the graph of the function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is a vertical shrink by a factor of [tex]\(\frac{1}{4}\)[/tex]. This means that every y-value of the parent function is scaled down to [tex]\(\frac{1}{4}\)[/tex] of its original value.
### Final Answer:
The transformation from the parent quadratic function [tex]\( y = x^2 \)[/tex] to the function [tex]\( y = \frac{1}{4}x^2 \)[/tex] is a vertical shrink by a factor of [tex]\(\frac{1}{4}\)[/tex]. This reduces the height of each point on the parabola by 75%, resulting in a wider and shorter parabola compared to the original.
