To determine what transformation is applied to the parent function [tex]\( f(x) \)[/tex] to get [tex]\( -2 f(x) \)[/tex], let us analyze each component of the transformation step-by-step.
1. Reflection in the [tex]\( x \)[/tex]-axis:
- When a negative sign is placed in front of [tex]\( f(x) \)[/tex], as in [tex]\( -f(x) \)[/tex], it implies a reflection of the graph of [tex]\( f(x) \)[/tex] across the [tex]\( x \)[/tex]-axis. This transformation mirrors all points of the graph over the [tex]\( x \)[/tex]-axis. Mathematically, [tex]\( -f(x) \)[/tex] flips the output values (or [tex]\( y \)[/tex]-values) of the function.
2. Vertical Stretch by a Factor of 2:
- When the function [tex]\( f(x) \)[/tex] is multiplied by a factor greater than 1, as in [tex]\( 2f(x) \)[/tex], it results in a vertical stretch of the graph. Specifically, multiplying [tex]\( f(x) \)[/tex] by 2 will stretch the output values of the function away from the [tex]\( x \)[/tex]-axis by a factor of 2. This means all [tex]\( y \)[/tex]-values (or output values) will be doubled.
Combining these two transformations for [tex]\( -2 f(x) \)[/tex], we get:
- The function [tex]\( f(x) \)[/tex] is first stretched vertically by a factor of 2, doubling all [tex]\( y \)[/tex]-values.
- Then, the resulting graph is reflected over the [tex]\( x \)[/tex]-axis, inverting the signs of all [tex]\( y \)[/tex]-values.
Hence, the transformations applied to the parent function [tex]\( f(x) \)[/tex] to get [tex]\( -2 f(x) \)[/tex] are:
- A reflection in the [tex]\( x \)[/tex]-axis.
- A vertical stretch by a factor of 2.
Therefore, the correct answer is:
D. a reflection in the [tex]\( x \)[/tex]-axis and a vertical stretch by 2
