The function F(x) = 2|x + 6| − 1 is translated 7 units down and 1 unit right. Write the equation of the new function.

Step-by-step solution

1. 1

   Locate the starting vertex. An absolute-value graph $a|x-p|+q$ has its corner at $(p,q)$. Writing $2|x+6|-1$ as $2|x-(-6)|+(-1)$ shows the vertex is at $(-6,-1)$ with a vertical stretch of factor $2$. Tracking the vertex is the quickest way to check the final answer.

2. 2

   Apply the horizontal shift, and mind the direction. Moving a graph $1$ unit to the right replaces every $x$ by $x-1$ — the counter-intuitive step, because the sign inside looks backwards:

   $2\bigl|(x-1)+6\bigr|-1=2|x+5|-1.$

   The vertex moves from $(-6,-1)$ to $(-5,-1)$, one unit right, confirming the substitution was made the right way round.

3. 3

   Apply the vertical shift. Moving $7$ units down subtracts $7$ from the whole function — this one acts outside the absolute value and does behave as expected:

   $2|x+5|-1-7=2|x+5|-8.$

4. 4

   Confirm with the vertex. The corner is now at $(-5,-8)$: one unit right and seven units down from $(-6,-1)$, exactly as required. The stretch factor $2$ is untouched, since translations never change the shape or the slopes of the two rays, which remain $\pm2$.

5. 5

   Spot-check a point. On the original graph, $F(-6)=2(0)-1=-1$. The corresponding point on the image should be $(-6+1,\,-1-7)=(-5,-8)$. Evaluating the answer: $2|-5+5|-8=-8$ ✓. A second check at $x=-3$: the answer gives $2(2)-8=-4$, and the pre-image point is $x=-4$ with $F(-4)=2(2)-1=3$, and $3-7=-4$ ✓.