Math Help - Graph of the equation is symmetric

I need help solving this problem, can anyone do a step by step guide on how to do this?

Determine whether the graph of the equation is symmetric with respect to the line Y = -X

1) Y = 2X - 1

One way of checking it is by slopes, because both x = -y and y = 2x +1 are straight lines only.
If y = 2x +1 were symmetrical with respect to y = -x, then y = 2x +1 should be perpendicular to y = -x. If not, then there is no symmetry.

y = -x
slope, m1 = -1

y = 2x +1
slope, m2 = 2

To be perpendicular, their slopes must be the negative reciprocals.
m2 = -1/m1
2 = -1 / -1
2 = 1
False, so the two lines are not perpendicular, and so y = 2x +1 is not symmetrical wirh respect to y = -x.

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Another way.

y = -x -------------axis of symmetry.

y = 2x +1 --------(i)

If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).

-x = 2(-y) +1
-x = -2y +1
2y = x +1
y = (1/2)(x +1) -------not the same as (i), so, no symmetry.

Originally Posted by ticbol

y = -x -------------axis of symmetry.

y = 2x +1 --------(i)

If (i) is symmetrical, when we swap the x and y into (i), we should get an equation similar or equal to (i).

-x = 2(-y) +1
-x = -2y +1

How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?

Originally Posted by Brazuca

How come the -x remained as a -x. While the 2y turned into a -2y and the -1 turned into a 1?

what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.

Originally Posted by Jhevon

what are you talking about? he replaced y with -x and x with -y and solved for y. that's it.

I don't understand.

If Y = -X then wouldn't the -X have to turn into a X for the Y to turn into a -Y?

-x = 2(-y) +1
-x = -2y +1

Wouldn't the problem have to turn into this?
-x = 2y + 1
x = 2(-y) +1
x = -2y + 1

Originally Posted by Brazuca

I don't understand.

we have
which is also saying that , this is line (1)

then we are given ........line (2)

to find if is symmetric with respect to line one, plug in and as line (1) directed. if we simplify and get the original formula for line (2), then line (2) is symmetric with respect to line (1)



Plug in and , we get:



solving for , we get:



which is not the original formula for line (2), so line (2) is NOT symmetric with respect to line (1)

Originally Posted by Jhevon

we have
which is also saying that , this is line (1)

Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.

I always looked at it as y = -x then -x = y. I was not switching the signs.

Originally Posted by Brazuca

Thanks, I had to read everything you posted step by step to understand that y = -x then x = -y.

I always looked at it as y = -x then -x = y. I was not switching the signs.

if you multiply both sides of the equation y = -x by -1 you get -y = x