Question

use transformations of the absolute value function, $f(x)=|x|$, to graph the function $g(x)=-2|x-1|+6$.apply.□ a. reflection about the x-axis□ b. horizontal stretch by a factor of 2□ c. horizontal shift 1 unit to the left□ d. horizontal shift 1 unit to the right□ e. vertical stretch by a factor of 2□ f. vertical shift 6 units up

Explanation:

Step1: Identify reflection

The negative sign outside $|x-1|$ reflects $f(x)$ over the x-axis: $-|x|$.

Step2: Identify vertical stretch

The coefficient 2 outside $|x-1|$ vertically stretches by factor 2: $-2|x|$.

Step3: Identify horizontal shift

The $x-1$ inside the absolute value shifts $f(x)$ right 1 unit: $-2|x-1|$.

Step4: Identify vertical shift

The $+6$ outside shifts the function up 6 units: $-2|x-1|+6$.

Answer:

A. Reflection about the x-axis
D. Horizontal shift 1 unit to the right
E. Vertical stretch by a factor of 2
F. Vertical shift 6 units up

To graph $g(x) = -2|x-1|+6$:

1. Start with $f(x)=|x|$, a V-shape with vertex at $(0,0)$.
2. Reflect over the x-axis to get $-|x|$.
3. Vertically stretch by a factor of 2 to get $-2|x|$.
4. Shift 1 unit right to get $-2|x-1|$.
5. Shift 6 units up to get $-2|x-1|+6$, with vertex at $(1,6)$, opening downward, with slopes $\pm2$.