Question 1118777: The graphs of y = -|x-a|+b and y = |x-c|+ d intersect at points (2 , 5)
and (8, 3 ). Find a+c Found 2 solutions by greenestamps, ikleyn:Answer by greenestamps(13200) (Show Source):
The slopes of the branches of the graphs of both functions are 1 and -1.
So the equations of the branches of each function that intersect at (2,5) are
(1) y = x+3 and (2) y = -x+7
And the equations of the branches of each function that intersect at (8,3) are
(3) y = x-5 and (4) y = -x+11
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.
So for one of the functions
The vertex of this graph is (4,7); the equation is y = -|x-4|+7.
For the other function
The vertex of this one is (6,1); the equation is y = |x-6|+1.
The values of a and c are 4 and 6; a+c = 4+6 = 10.
Here is a graph....
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.
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.
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.
It is a "joke" problem for advanced students.
It does not assume that you will restore the functions and solve equations.
But it assumes that you firmly know that
- the first function has a vertex at the point (a,b) with two branches going down perpendicularly and making angles 45 degrees with the axes;
- the second function has a vertex at the point (c,d) with two branches going up perpendicularly and making angles 45 degrees with the axes.
Then you have a rectangle as the intersection of all the branches (I refer to the solution and the plot / (a sketch is just enough !)
by @greenestamps), and the problem asks about a+c.
Notice that is x-coordinate of the mid-point of one diagonal of the rectangle, and it is the same as x-coordinate
of the mid-point of the other diagonal = 5.
Thus = 5 and, hence, a+c = 10.
Solved.
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This solution is parallel to the final part of the @greenestamps solution,
but presented in other words/terms to provide better understanding for you.
The major idea of the problem and the major idea of the solution is in that
it does not require any calculations (except the absolute minimum).
