Final answer:
To create a wider graph that shifts right by 3 units and up by 8 units from the equation y = 1/2|x + 2| - 6, the new equation is y = 1/4|x - 1| + 2, which includes adjustments for width, horizontal shift, and vertical shift.
Step-by-step explanation:
To create a new equation that is wider, shifts right 3 units, and shifts up 8 units from the original equation y = 1/2|x + 2| - 6, we need to adjust the absolute value expression, horizontal shift, and vertical shift.
First, to make the new graph wider, we can decrease the multiplier in front of the absolute value expression. Since the original equation has a multiplier of 1/2, we could use a smaller fraction such as 1/4. This makes the new equation's rate of increase or decrease half as steep, effectively making it wider.
Next, to shift the graph to the right by 3 units, we add 3 to the x-value within the absolute value expression. So the expression |x + 2| becomes |x - 1| because (x + 2) - 3 simplifies to (x - 1).
Finally, to shift the graph upwards by 8 units, we add 8 to the entire equation. The -6 at the end of the original equation becomes +2 when 8 units are added.
Combining these changes, the new equation is y = 1/4|x - 1| + 2.