Question

use transformations of the absolute value function, $f(x)=|x|$, to graph the function $g(x)= -2|x-1|+6$.□ 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 upuse the graphing tool to graph the equation.click to enlarge

Explanation:

Step1: Identify base function

Base function: $f(x)=|x|$

Step2: Apply horizontal shift

Shift right 1 unit: $|x-1|$

Step3: Apply vertical stretch/reflection

Stretch by 2, reflect over x-axis: $-2|x-1|$

Step4: Apply vertical shift

Shift up 6 units: $g(x)=-2|x-1|+6$

Step5: List correct transformations

Match to given options.

Answer:

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 $f(x)=|x|$ (V-shape with vertex at (0,0)).
2. Shift the graph 1 unit to the right (vertex moves to (1,0)).
3. Stretch vertically by a factor of 2 (steeper sides, vertex remains (1,0)).
4. Reflect the graph over the x-axis (V-shape opens downward).
5. Shift the entire graph 6 units up (final vertex at (1,6)).