To determine what transformation occurs when the absolute value function [tex]\( f(x) = |x| \)[/tex] is multiplied by [tex]\(-3\)[/tex], let's analyze the steps:
1. Starting Function: Begin with the given function [tex]\( f(x) = |x| \)[/tex].
2. Apply the Multiplication by [tex]\(-3\)[/tex]: When we multiply [tex]\( f(x) \)[/tex] by [tex]\(-3\)[/tex], the new function becomes [tex]\( g(x) = -3|x| \)[/tex].
Next, we look at what this transformation does to the graph:
- Reflection across the [tex]\( x \)[/tex]-axis:
- The original function [tex]\( f(x) = |x| \)[/tex] has all its values non-negative. However, multiplying by [tex]\(-3\)[/tex] changes the sign of all values, making them non-positive. This effect is equivalent to flipping the graph of [tex]\( f(x) = |x| \)[/tex] upside down, reflecting it across the [tex]\( x \)[/tex]-axis.
- Vertical Stretch by a Factor of 3:
- The multiplication by 3 (regardless of the negative sign) scales the values of [tex]\( |x| \)[/tex] by a factor of 3. This means that every point on the graph is moved three times farther from the [tex]\( x \)[/tex]-axis, which is a vertical stretch by a factor of 3.
Therefore, when the absolute value function [tex]\( f(x)=|x| \)[/tex] is multiplied by [tex]\(-3 \)[/tex], the graph undergoes:
1. A reflection across the [tex]\( x \)[/tex]-axis.
2. A vertical stretch by a factor of 3.
So, the correct transformation is:
Reflection across the [tex]\( x \)[/tex]-axis and vertical stretch by 3.
