Question

use transformations of the absolute value function, f(x) = |x|, to graph the function g(x) = -2|x + 1| - 5.

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 shift 1 unit to the right
b. reflection about the x-axis
c. vertical stretch by a factor of 2
d. horizontal stretch by a factor of 2
e. horizontal shift 1 unit to the left
f. vertical shift 5 units down

Explanation:

Step1: Analyze horizontal shift

The parent function is \( f(x) = |x| \), and the transformed function has \( |x + 1| \). For horizontal shifts, the form \( |x - h| \) shifts \( h \) units. Here, \( h=-1 \), so it's a shift 1 unit left (since \( h=-1 \) means \( x - (-1)=x + 1 \)), so option E is correct, A is wrong.

Step2: Analyze reflection

The function has a negative sign in front of the absolute value: \( -2|x + 1| \). A negative sign in front of the function reflects it over the x - axis, so option B is correct.

Step3: Analyze vertical stretch

The coefficient 2 in front of the absolute value (after reflection) is a vertical stretch. The general form for vertical stretch is \( a|x| \) where \( |a|>1 \) is a stretch. Here \( a = 2 \), so vertical stretch by factor 2, option C is correct, D is wrong (horizontal stretch would be with a coefficient inside the absolute value affecting x, not outside).

Step4: Analyze vertical shift

The function has \( - 5 \) at the end. This is a vertical shift down 5 units, so option F is correct.

Answer:

B. Reflection about the x - axis, C. Vertical stretch by a factor of 2, E. Horizontal shift 1 unit to the left, F. Vertical shift 5 units down