Synmetrical graphs

Question

Consider a gaph of the function y=2x^2 - 6x + 2 in xy-plane.Next, consider another graph that is symmetrical to the previous graph with respect to the line x=2. The graph that is symmetrical to the latter graph with respect to the line y=3 is described as:

y= ax2+bx+c

what will be a,b and c?

Answer 1

y= 2x^2 -6x+2 = 2(x^2-3x ) +2

complete the square, add & subtract 9/8

y= 2(x^2 -3x + 9/4) + 2- 9/2

y =2(x-3/2)^2 - 5/2

vertex = (3/2, -5/2)

symmetric about x=2

3/2 is 1/2 to the left of 2, 5/2 is 1/2 to the right of 2

means the vertex is now (5/2, -5/2)

and the equation is now

y =-2(x-5/2)^2 -5/2

= 2(x^2 -5x + 25/4) -5/2

= 2x^2 -10x +25/2 -5/2

= 2x^2 -10x +10

= 2(x^2-5x) +10

next wrinkle in the problem is finding the equation symmetric about y=3

so

find a y value equidistant from 3 as 10 is.

10 is 7 above 3, 7 below 3 is -4

also flip the parabola upside down, change the sign of a, change 2 to =-2

the equation is then

y=-2x^2+10x -4

a=-2, b=10 c= -4