Certainly! Let's examine the transformation from the function [tex]\( f(x) = x^2 \)[/tex] to the function [tex]\( g(x) = -4x \)[/tex].
1. Understanding [tex]\( f(x) = x^2 \)[/tex] and [tex]\( g(x) = -4x \)[/tex]:
- [tex]\( f(x) = x^2 \)[/tex] is a quadratic function. Its graph is a parabola that opens upwards and is symmetric about the y-axis.
- [tex]\( g(x) = -4x \)[/tex] is a linear function. Its graph is a straight line with a negative slope of -4.
2. Reflection:
- Reflecting [tex]\( f(x) = x^2 \)[/tex] about the x-axis would change its form to [tex]\( f(x) = -x^2 \)[/tex]. This transformation flips the parabola so that it opens downwards.
3. Scaling and Change of Function Type:
- To get from a quadratic function [tex]\( f(x) = -x^2 \)[/tex] to a linear function [tex]\( g(x) = -4x \)[/tex], there is a vertical and horizontal scaling involved, combined with a change from a quadratic to a linear function. Essentially, the quadratic nature of the function changes, simplifying it into a linear function.
4. Combining the Transformations:
- The overall transformation involves initially reflecting the original quadratic function across the x-axis and then transforming it into a linear function with a slope of -4.
Therefore, the described transformation from [tex]\( f(x) = x^2 \)[/tex] to [tex]\( g(x) = -4x \)[/tex] includes a reflection about the x-axis and a change from a quadratic function to a linear function.
Hence, the correct description of the transformation is a reflection and a change from quadratic to linear.
