Question

use transformations of the absolute value function, ( f(x) = |x| ), to graph the function ( g(x) = -3|x - 1| - 2 ).

what transformations are needed in order to obtain the graph of ( g(x) ) from the graph of ( f(x) )?
select all that apply.

a. horizontal stretch by a factor of 3
b. horizontal shift 1 unit to the right
c. horizontal shift 1 unit to the left
d. vertical stretch by a factor of 3
e. reflection about the x - axis
f. vertical shift 2 units down

use the graphing tool to graph the equation.

Explanation:

To determine the transformations from \( f(x) = |x| \) to \( g(x) = -3|x - 1| - 2 \), we analyze each part:

• The \( (x - 1) \) inside the absolute value indicates a horizontal shift. For \( |x - h| \), \( h>0 \) shifts right \( h \) units, so \( h = 1 \) means a horizontal shift 1 unit to the right (matches option B).
• The coefficient \( -3 \) in front: the negative sign reflects the graph about the \( x \)-axis (option E), and the factor of 3 is a vertical stretch (since it's outside the absolute value, affecting the \( y \)-values, so vertical stretch by factor 3, option D).
• The \( -2 \) at the end is a vertical shift. For \( f(x)+k \), \( k<0 \) shifts down \( |k| \) units, so \( k=-2 \) means a vertical shift 2 units down (option F).
• Option A is incorrect because a horizontal stretch would be of the form \( |\frac{1}{a}x| \), not a factor of 3 outside the horizontal shift. Option C is incorrect as \( (x - 1) \) shifts right, not left.

Answer:

B. Horizontal shift 1 unit to the right, D. Vertical stretch by a factor of 3, E. Reflection about the x - axis, F. Vertical shift 2 units down