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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 \(f(x) = 1.8\)?

 

Guest May 12, 2018
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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

 

 

 

cool cool cool

CPhill  May 12, 2018

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