Question

use transformations of the absolute value function, $f(x)=|x|$, to graph the function $g(x)=-4|x-1|-5$.apply.□ a. horizontal shift 1 unit to the right□ b. horizontal shift 1 unit to the left□ c. horizontal stretch by a factor of 4□ d. reflection about the x-axis□ e. vertical stretch by a factor of 4□ f. vertical shift 5 units down

Explanation:

Step1: Identify horizontal shift

For $g(x) = -4|x-1| -5$, compare to $f(x-h)$: $h=1$, so shift right 1 unit.

Step2: Identify reflection/stretch

The coefficient $-4$ outside $|x-1|$: negative sign reflects over x-axis, $|4|$ is vertical stretch by 4.

Step3: Identify vertical shift

The $-5$ at the end is a vertical shift down 5 units.

Answer:

A. Horizontal shift 1 unit to the right
D. Reflection about the x-axis
E. Vertical stretch by a factor of 4
F. Vertical shift 5 units down

To graph $g(x)$:

1. Start with $f(x)=|x|$, shift all points 1 unit right.
2. Reflect the resulting graph over the x-axis.
3. Vertically stretch the reflected graph by a factor of 4 (multiply y-values by 4).
4. Shift the entire graph 5 units down.

The vertex of $g(x)$ will be at $(1, -5)$, opening downward with a steeper slope than $f(x)$.