Question

What is the effect on the graph of f(X) = |X| when the function is changed to

g(X) = -2|x|?
OA. The graph is reflected across the x-axis and stretched vertically by
a factor of 2.
O B. The graph is shifted to the left 2 units.
C. The graph is shifted down 2 units.
D. The graph is reflected across the x-axis and compressed vertically
by a factor of 2.

Answer 1

A. The graph is reflected across the x-axis and stretched vertically by

a factor of 2.

Explanation:

Transformations

For a > 0

`f(x+a) \implies f(x) \: \textsf{translated $a$ units left}.`

`f(x-a) \implies f(x) \: \textsf{translated $a$ units right}.`

`f(x)+a \implies f(x) \: \textsf{translated $a$ units up}.`

`f(x)-a \implies f(x) \: \textsf{translated $a$ units down}.`

`a\:f(x) \implies f(x) \: \textsf{stretched parallel to the $y$-axis (vertically) by a factor of $a$}.`

`f(ax) \implies f(x) \: \textsf{stretched parallel to the $x$-axis (horizontally) by a factor of $(1)/(a)$}.`

`-f(x) \implies f(x) \: \textsf{reflected in the $x$-axis}.`

`f(-x) \implies f(x) \: \textsf{reflected in the $y$-axis}.`

Given functions:

`f(x)=|x|`

`g(x)=-2|x|`

The series of transformations that take function f(x) to function g(x) are:

1. Reflection across the x-axis:

`-f(x)\implies g(x)-|x|`

2. Vertical stretch by a factor of 2:

`-2f(x) \implies g(x)=-2|x|`