Length of thread is 2/7m long. mr. keller used 2/5 of it. how much thread did he have left?

The parent function is clearly y=|x|, which has the shape of a " V ", with vertex at the origin, and whose "wing" open at 45 degrees. The axis of symmetry is the y-axis.

Let's consider the effects of each of 10, -2, -7 on the graph.

i) From y=|x| to y=|10x|.

We note that in the parent function we have the points (1, 1) and (-1, 1), while in the new function we have (1, 10) and (-1, -10). This means that the graph has been stretched vertically by a factor of 10.

However, neither the vertex, nor the axis of symmetry have moved.

ii) From y=|10x| to y=|10x-2|. Now we want to check at what values of x does the new function take the value 10. So we solve:

|10x-2|=10

This means that 10x-2=10 or 10x-2=-10. Thus:

10x=12 or 10x=-8.

Finally, we see that x=12/10=1.2 or x=-8/10=-0.8. So, instead of (-1, 10) and (1, 10) we now have:

(-0.8, 10) and (1.2, 10). This helps us see that the function has been shifted to the right by 0.2. (This comes from -2/10 in |10x-2|.) 

Thus, the new vertex is at (0.2, 0) and axis of symmetry is the vertical line x=0.2.

iii) From y=|10x-2| to y=|10x-2|-7. This is a shift down by 7 units of the whole graph.

Thus, to conclude:

The vertex is at (0.2, -7); The axis of symmetry is the vertical line x=0.2; The graph has been stretched vertically by a factor of 10.