Let's analyze the transformations applied to the function [tex]\( f(x) = x^2 \)[/tex] to transform it into [tex]\( g(x) = -3x^2 + 6x - 60 \)[/tex].
1. Getting an overview of [tex]\( g(x) = -3x^2 + 6x - 60 \)[/tex] compared to [tex]\( f(x) = x^2 \)[/tex]:
- The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( g(x) \)[/tex] is [tex]\(-3\)[/tex]. This indicates that the parabola opens downwards (since the coefficient is negative) and is made narrower compared to the parabola [tex]\( f(x) = x^2 \)[/tex] (since [tex]\(|-3| > 1\)[/tex]).
- Terms involving [tex]\( x \)[/tex] can suggest potential horizontal shifts or other transformations.
- The constant term [tex]\(-60\)[/tex] suggests a vertical shift.
2. Analyzing specific transformations:
- Narrowing of the graph: The coefficient of [tex]\( x^2 \)[/tex] in [tex]\( g(x) \)[/tex] is [tex]\(-3\)[/tex], which means the graph is narrower than [tex]\( f(x) = x^2 \)[/tex].
- Reflection over the x-axis: The negative sign in front of the [tex]\( -3x^2 \)[/tex] term suggests that the graph is reflected over the x-axis. Note that this creates an opening downwards as opposed to upwards.
- Vertical shift: The constant term [tex]\(-60\)[/tex] indicates that the graph is shifted down by 60 units compared to the original graph of [tex]\( f(x) = x^2 \)[/tex].
Therefore, considering the question,
- "The graph of [tex]\( f(x) = x^2 \)[/tex] is made narrower." is accurate as one of the transformations.
- "The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted right 6 units." is not correct based on the transformations observed.
- "The graph of [tex]\( f(x) = x^2 \)[/tex] is shifted down by 48 units." is also not correct as the shift is by 60 units.
- "The graph of [tex]\( f(x) = x^2 \)[/tex] is reflected over the [tex]\( y \)[/tex]-axis." is not correct as the reflection is over the x-axis due to the negative coefficient.
Thus, one of the transformations applied is:
- The graph of [tex]\( f(x) = x^2 \)[/tex] is made narrower.
