Answer :
To determine which function [tex]\( g(x) \)[/tex] is a transformation of the quadratic parent function [tex]\( f(x) = x^2 \)[/tex], we need to analyze how each option represents a transformation. The parent function [tex]\( f(x) = x^2 \)[/tex] can be transformed through vertical stretches, vertical shrinks, and reflections. Let's analyze each option in detail:
### Option A: [tex]\( g(x) = 3x^2 \)[/tex]
This transformation can be understood as follows:
- Vertical Stretch: The coefficient 3 in front of [tex]\( x^2 \)[/tex] means that for any given [tex]\( x \)[/tex], the value of [tex]\( g(x) \)[/tex] will be three times the value of [tex]\( f(x) \)[/tex]. Therefore, the graph is stretched vertically by a factor of 3.
### Option B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
This transformation involves:
- Vertical Shrink: The coefficient [tex]\(-\frac{1}{3}\)[/tex] indicates that the function is vertically shrunk by a factor of [tex]\( \frac{1}{3} \)[/tex]. This means the graph is compressed towards the x-axis.
- Reflection: The negative sign indicates a reflection across the x-axis. Thus, it flips the graph upside down.
### Option C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
This transformation can be described as:
- Vertical Shrink: The coefficient [tex]\( \frac{1}{3} \)[/tex] means the function is vertically compressed by a factor of [tex]\( \frac{1}{3} \)[/tex]. The graph is closer to the x-axis by this fractional factor.
### Option D: [tex]\( g(x) = -3x^2 \)[/tex]
Here, the transformation involves:
- Vertical Stretch: Similar to Option A, the coefficient 3 indicates the graph is stretched vertically by a factor of 3.
- Reflection: The negative sign indicates a reflection across the x-axis, flipping the graph upside down.
To summarize, the original question pertains to identifying how the function [tex]\( g(x) \)[/tex] transforms the parent function [tex]\( f(x) = x^2 \)[/tex] considering vertical stretches, vertical shrinks, and reflections. Thus, the transformations can be represented as follows:
- Option A: Vertical stretch by a factor of 3.
- Option B: Vertical shrink by a factor of [tex]\( \frac{1}{3} \)[/tex] and reflection across the x-axis.
- Option C: Vertical shrink by a factor of [tex]\( \frac{1}{3} \)[/tex].
- Option D: Vertical stretch by a factor of 3 and reflection across the x-axis.
These transformations confirm that each option represents a distinct type of transformation of the parent function [tex]\( f(x) = x^2 \)[/tex].
### Option A: [tex]\( g(x) = 3x^2 \)[/tex]
This transformation can be understood as follows:
- Vertical Stretch: The coefficient 3 in front of [tex]\( x^2 \)[/tex] means that for any given [tex]\( x \)[/tex], the value of [tex]\( g(x) \)[/tex] will be three times the value of [tex]\( f(x) \)[/tex]. Therefore, the graph is stretched vertically by a factor of 3.
### Option B: [tex]\( g(x) = -\frac{1}{3}x^2 \)[/tex]
This transformation involves:
- Vertical Shrink: The coefficient [tex]\(-\frac{1}{3}\)[/tex] indicates that the function is vertically shrunk by a factor of [tex]\( \frac{1}{3} \)[/tex]. This means the graph is compressed towards the x-axis.
- Reflection: The negative sign indicates a reflection across the x-axis. Thus, it flips the graph upside down.
### Option C: [tex]\( g(x) = \frac{1}{3}x^2 \)[/tex]
This transformation can be described as:
- Vertical Shrink: The coefficient [tex]\( \frac{1}{3} \)[/tex] means the function is vertically compressed by a factor of [tex]\( \frac{1}{3} \)[/tex]. The graph is closer to the x-axis by this fractional factor.
### Option D: [tex]\( g(x) = -3x^2 \)[/tex]
Here, the transformation involves:
- Vertical Stretch: Similar to Option A, the coefficient 3 indicates the graph is stretched vertically by a factor of 3.
- Reflection: The negative sign indicates a reflection across the x-axis, flipping the graph upside down.
To summarize, the original question pertains to identifying how the function [tex]\( g(x) \)[/tex] transforms the parent function [tex]\( f(x) = x^2 \)[/tex] considering vertical stretches, vertical shrinks, and reflections. Thus, the transformations can be represented as follows:
- Option A: Vertical stretch by a factor of 3.
- Option B: Vertical shrink by a factor of [tex]\( \frac{1}{3} \)[/tex] and reflection across the x-axis.
- Option C: Vertical shrink by a factor of [tex]\( \frac{1}{3} \)[/tex].
- Option D: Vertical stretch by a factor of 3 and reflection across the x-axis.
These transformations confirm that each option represents a distinct type of transformation of the parent function [tex]\( f(x) = x^2 \)[/tex].
