Question 1118777
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The slopes of the branches of the graphs of both functions are 1 and -1.<br>
So the equations of the branches of each function that intersect at (2,5) are
(1) y = x+3  and  (2) y = -x+7<br>
And the equations of the branches of each function that intersect at (8,3) are
(3) y = x-5  and  (4) y = -x+11<br>
Equations (1) and (4) are the two branches of the graph of one of the functions; equations (2) and (3) are the two branches of the graph of the other.<br>
So for one of the functions
{{{x+3 = -x+11}}}
{{{2x = 8}}}
{{{x = 4}}}
{{{y = x+3 = 7}}}
The vertex of this graph is (4,7); the equation is y = -|x-4|+7.<br>
For the other function
{{{-x+7 = x-5}}}
{{{2x = 12}}}
{{{x = 6}}}
{{{y = x-5 = 1}}}
The vertex of this one is (6,1); the equation is y = |x-6|+1.<br>
The values of a and c are 4 and 6; a+c = 4+6 = 10.<br>
Here is a graph....<br>
{{{graph(400,400,-2,10,-2,10,-abs(x-4)+7,abs(x-6)+1)}}}<br>
Note that the question asked only for the sum of a and c; those are the x coordinates of the vertices of the two functions.<br>
By looking at the graph, it might be apparent to you that, in problems like this, that sum is always going to be just the sum of the x coordinates of the two points of intersection.<br>
If you see that, then the next time you see a problem like this you can write down the answer without doing any work.  The answer is the sum of the x coordinates of the two points of intersection: 2+8=10.