The complete graph of \(y=f(x)\), which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is \(1\).)
What is the sum of the \(x\)-coordinates of all points where \(f(x) = 1.8\)?
+128406 If we let f(x) = y, we are looking for the the x coordinates of the form ( x , 1.8)
The first segment which contains a points like this is the segment from (0, 0) to (1,2)
The slope of the line containing this segment is [ 2 - 0 ] / [1 - 0] = 2
And the equation of this line is y = 2x
So...when y = 1.8, we have
1.8 = 2 x divide both sides by 2
.9 = x
So....the first point is (.9, 1.8)
The next segment containing the y valu of 1.8 is the segment from (1,2) to (2,1)
The slope of the line containing this segment is (1 - 2) /( 2 -1) = -1/1 = -1
And the equation of this line is
y = -(x -1) + 2
y = -x + 3
And when y = 1.8, we have
1.8 = -x + 3 subtract 3 from both sides
-1.2 = -x
1.2 = x
So.....the second point is ( 1.2, 1.8)
The last point where y = 1.8 is on the segment from (2,1) to (3,3)
The sloe of the line containing this point is (3 - 1) /( 3 - 2) = 2/ 1 = 2
And the equation of the line containing this segment is
y =2 ( x -3) + 3
y = 2x - 6 + 3
y = 2x - 3
And when y = 1.8, we have
1.8 = 2x - 3 add 3 to both sides
4.8 = 2x divide both sides by 2
2.4 = x
So....the last point is ( 2.4, 1.8)
So....the sum of the x coordinates where f(x) = 1.8 is .9 + 1.2 + 2.4 = 4.5
