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The graph of the parabola defined by the equation y=(x-2)^2 + 3 is rotated 180 degrees about its vertex, then shifted 3 units to the left, then shifted 2 units down. The resulting parabola has zeros at x=a and x=b. What is a+b?

 

Help me solve this please! Thanks

Guest Jan 4, 2018
edited by Guest  Jan 4, 2018
 #1
avatar+7153 
+1

y  =  (x - 2)^2 + 3

 

This parabola has its vertex at  (2, 3) . Rotating it  180°  about its vertex has the same effect as flipping it over the x-axis and then shifting it up  6  units. So our new parabola has the equation...

 

y  =  -[ (x - 2)^2 + 3 ] + 6

 

Now let's shift it  3  units to the left...

 

y  =  -[ (x - 2 + 3)^2 + 3 ] + 6

 

And  2  units down...

 

y  =  -[ (x - 2 + 3)^2 + 3 ] + 6 - 2          Then simplify.

 

y  =  -(x +1)^2 + 1

 

The zeros of this parabola are the  x  values when...

 

0  =  -(x +1)^2 + 1

 

(x +1)^2  =  1

 

x + 1   =   ±√1

 

x + 1  =  1          or         x + 1  =  -1

x  =  0                or         x  =  -2

 

And the sum of the zeros   =   0 + -2   =   -2

 

Here's a graph of the original parabola and the transformed parabola.

hectictar  Jan 4, 2018
edited by hectictar  Jan 4, 2018
edited by hectictar  Jan 4, 2018
 #2
avatar+87294 
+2

y = (x - 2)^2  + 3

 

If this is rotated 180° about its vertex, the vertex does not change..........the new function is

 

y  =  - (x- 2)^2  + 3

 

Shifting this to the left 3 units left and 2 units down  results in the function

 

y  =  - (x + 1)^2   +  1

 

To find the zeroes, we have

 

-(x + 1)^2  +  1  =  0

 

-(x +1)^2  =  -1

 

(x + 1)^2  =  1

 

x + 1  =  ±1

 

So

 

x +  1  =  1 ⇒  x  = 0

 

x + 1  =  - 1  ⇒  x  = -2

 

So    a  +  b  =    0   +   -2    =    - 2

 

Here's the graph :  https://www.desmos.com/calculator/zbkwtqadyr

 

 

cool cool cool

CPhill  Jan 4, 2018

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